## June 30, 2009

### Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

#### Posted by John Baez

The ‘big three’ science publishers — Elsevier, Springer, and Wiley–Blackwell — like to argue that their high prices pay for high quality. Recent events cast a tinge of doubt on this. We all know about the case of El Naschie, and Elsevier’s fake medical journals. Now Springer has published a book that purports to contain elementary proofs of Fermat’s Last Theorem and Goldbach’s Conjecture!

Here it is:

The author recently posted about his book on the newsgroup sci.math.research. Here is the post, with a comment by the newgroup’s moderator:

Please notice that my book “Associative Digital Network Theory” was just released by Springer Verlag, see:

http://www.springer.com/computer/communications/book/978-1-4020-9828-4

It is about the use of function composition (semigroup theory) as binding principle for the three main levels of functions applied in computer engineering : State-machines, Arithmetic and (Boolean) logic, corresponding to a hiërarchy of associative algebra’s : non-commutative, commutative, and idempotent,respectively : $a(b c) = (a b ) c =a b c, a b = b a , a a = a$.

In Chapter 2 the five basic state machines are derived, and the decomposition of permutation machines with a non-trivial simple group as closure is in Ch.3 (e.g. group A5 of order 60 as coupled network of cyclic groups of order 2 [twice] , 3, 5). Ch.4 discusses the general decompositon of finite semigroups / statemachines, as (possibly coupled) network of the five basic statemachine types, including the two non-commutative memory types: branch- and set/reset machines.

‘Planar Boolean Logic’ (Ch.5) is defined as practically related to symmetric Boolean functions, with a proof that all $BF_n$ of n<5 inputs are planar. Also various forms of fault tolerant logic designs are treated and compared.

Moreover, proofs of FLT and Goldbach’s Conjecture (Ch.8, 9) are given, both using a ‘residue-and-carry’ method (with proper choice of modulus), as well as a result of “Waring for residues” (Ch.10) for prime power moduli: each residue mod $p^k$ [$k>0$] is the sum of at most 4 p-th power residues.

Log-arithmetic (Ch.11) over double base 2 and 3 is discussed , using : each odd residue mod $2^k$ is a unique signed power of 3, as well as over single base 2 (a 32-bit VLSI implementation as an Euro ‘Esprit’ project).

—–

dr. Nico F. Benschop, Geldrop (NL) – Amspade Research —- nfbensc…@onsbrabantnet.nl

[mod note. Springer have clearly published this book by Benschop. Furthermore, this book claims to give elementary proofs of FLT and of Goldbach’s conjecture. Benschop has posted proofs of such things in the past—for example chapter 8 of this book appears to the moderator to be largely the same as the proof of Goldbach’s conjecture announced at

http://arxiv.org/abs/math/0103091

in 2001. As far as the moderator knows, the mathematical community has not yet accepted the proof described in this paper. Furthermore, viewers with access to math. reviews might want to see review number MR1831809, which appears to the moderator to pertain to another chapter of this book. The moderator wants to make it clear that acceptance of this sci.math.research post is nothing more than *acknowledgement of the statement that Springer has published the book*, which the moderator believes to be of independent interest, and does not imply that the moderator has read, or believes, any of the stronger claims made in the book.]

Posted at June 30, 2009 6:55 PM UTC

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### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Springer has also recently published

Farming Human Pathogens
Ecological Resilience
and Evolutionary Process

By Rodrick Wallace · Deborah Wallace ·
Robert G. Wallace

Have a look and tell me if the comparison between the two books is unfair.

Posted by: Eugene Lerman on June 30, 2009 8:36 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The link for this book is
here
There is also an online version available.

Posted by: Maarten Bergvelt on June 30, 2009 8:47 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Perhaps you can explain why this book should be compared to the one in the main post? It’s not immediately obvious to me what is so shocking about the things that it claims to show, though perhaps the application of information theory is slightly unexpected?

Posted by: Kenny Easwaran on July 1, 2009 10:05 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The reason for bringing up the Wallace, Wallace and Wallace book is that it reads to me like something the Bogdanov brothers could have written had they been doing math biology.

The point I am trying to make is that the Benschop book is not the only “controversial ” book in the Springer portfolio. I came across the Wallace et al book by accident: I was doing a Google search trying to find something on groupoid symmetries in dynamics.

I wonder how many other interesting book are out there.

Posted by: Eugene Lerman on July 1, 2009 4:59 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Just for the record, the central argument in Farming Human Pathogens, in chapters 2-4, is based on a long peer reviewed article:

R. Wallace and D. Wallace (2008), Punctuated equilibrium in statistical models of generalized coevolutionary resilinece…, Transactions on Computational Systems Biology IX, 23-85.

Chapter 5 and most of 6 are also based on published peer reviewed articles.

Posted by: Rodrick Wallace on July 1, 2009 7:28 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I don’t work anywhere close to mathematical biology, but I’ve been to a lot of talks on that field and at a superficial glance I don’t see anything that looks shocking or controversial about the Wallace book. (In particular there’s nothing unexpected about applying information theory. Groupoids are slightly unexpected, but applied mathematics can be surprisingly omnivorous.) The treatment of mathematics is mostly not up to pure mathematicians’ standards, but that’s to be expected.

In any case, as far as I can see they aren’t making any big, hard-to-believe claims.

Posted by: Mark Meckes on July 1, 2009 9:17 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

If the mathematical ideas are reasonable (and other posters have suggested that they are, although I wouldn't be able to tell you without reading the book or other such material), then this actually looks interesting.

Posted by: Toby Bartels on July 1, 2009 11:28 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Again, for the record.

[1] Jim Glazebrook has cleaned up and greatly extended the mathematics in Farming Human Pathogens:

J.F. Glazebrook and R. Wallace, (2009) Small worlds and Red Queens in the Global Workspace…, Cognitive Systems Research. Online doi:10.1016/j.cogsys.2009.01.002.

[2] A quite elementary and very clear biological application can be found in

R. Wallace (2009), Metabolic constraints on the eukaryotic transition, Origins of Life and Evolution of Biospheres, 39:165-176.

Using the methods of that paper and introducing perspectives from groupoid stereochemistry permits another approach to the biological homochirality conundrum. Ron Brown was kind enough to post a preprint about this on his Groupoid web page.

[3] One essential problem we try to address is how the genetic and cultural heritage systems of human populations interact. Taking a replicator dynamics/population genetics approach leads directly to cultural ‘memes’ that simply aren’t observed. Another attack is to back off and take a ‘weak’ perspective, treating genetic and cultural inheritance as information sources – generalized languages – constrained by the (hopefully robust) asymptotic limit theorems of information theory. The Feynman/Bennett duality between information and free energy leads to empirical Onsager relations, and ‘highly natural’ equivalence class structures lead to groupoids. A generalization of Landau’s spontaneous symmetry breaking arguments generates phase transition-like punctuation in complicated information source interactions.

The next step seems to be the highly nontrivial task of converting these probability models to statistical tools useful in data analysis.

Since 1998 we have published about 20 papers on these and related topics in peer reviewed journals. I have not found editors or reviewers unable to understand our work.

Oh well, I suppose the only really bad publicity is no publicity.

Anyone interested may contact me at

wallace@pi.cpmc.columbia.edu

Preprints/reprints should be sent to

rodrick.wallace@gmail.com

Posted by: Rodrick Wallace on July 2, 2009 5:15 PM | Permalink | Reply to this

### This margin’s too small; Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I can also generate “elementary proofs of Fermat’s Last Theorem and Goldbach’s Conjecture.” I simply make no claims that they are VALID. Hello, Earth to Springer, Earth to Springer…

Posted by: Jonathan Vos Post on June 30, 2009 10:42 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I wonder if the use of dieresis on “hiërarchy” is somehow correlated with the quality of the book.

Posted by: Tom Ellis on June 30, 2009 11:42 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

While some editor should have corrected this, the author might be forgiven this mistake as it is a logical mistake to make if you’re a native Dutch speaker (as this person seems to be). However, the author claims a PhD from Waterloo University, undermining my point.

The author seems to be some sort of electrical engineer, and also a frequent poster on sci.math. Looking at his webpage, one wonders what his crackpot score would be…

Posted by: Jan on July 1, 2009 8:51 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

How on earth did this happen? Have the editors at Springer lost their minds? Someone ought to be fired for this.

Even aside from the crank mathematics, take a look at page 18 (on the sample pages available for free). There’s an ASCII art diagram, together with eccentric typesetting like heavy use of underlining. If Springer did any copyediting or typesetting, then I shudder to think of what it must have looked like before. Or look at another ASCII art figure on page 22. Are we seriously paying 77 cents per page for ASCII art? How can Springer possibly justify such a price, especially when they appear to have put little or no effort into the book?

I would really like to know who approved this book and what sort of review process (if any) it underwent. We deserve an explanation, of either why they stand by their decision or how we can be confident that this will never happen again.

Posted by: Anonymous on July 1, 2009 5:16 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

“How can Springer possibly justify such a price, especially when they appear to have put little or no effort into the book?”

Well, if they behave just like Elsevier, then all the high rating and rave reviews their books recieve on the internet are perhaps something to do with it:

http://news.bbc.co.uk/1/hi/magazine/8118577.stm

Posted by: Ben on July 1, 2009 12:45 PM | Permalink | Reply to this

### Was a Wikipedia Spammer

I while back I noticed this guy was adding references to his work to all sorts of Wikipedia mathematical pages.

They seem to be removed now.

Posted by: RodMcGuire on July 1, 2009 7:01 AM | Permalink | Reply to this

### Re: Was a Wikipedia Spammer

Of course, now the guy’s claims are in print, they can be added back into Wikipedia along with a citation, making them satisfy the rules for not being deleted.

Posted by: Dan Piponi on July 1, 2009 10:16 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The author could have added a proof of the Riemann hypothesis, for good measure.

Posted by: GS on July 1, 2009 11:10 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Hi GS: It did cross my mind, having such a powerful tool as residue-and-carry (viz. applying semigroup structure to elementary arithmetic problems, and NOT forgetting about the carry - which was normally done by number theorists since Hensel… who’s extension lemma was a major block against progress - only a CS engineer familiar with the pain/importance of carries in arithmetic was apparently necessary to see that the Hensel-lift can be broken in special cases, one of which are the cubic roots of 1 mod p^k, the extension of FST that Fermat probably discovered soon after his FST mod p in 1637).

However, Riemann’s Hypothesis works in the complex plain, that is: on real-pairs, which is ‘too far from my bed’. Moreover, in industry (where I worked in the digital VLSI domain as an EE for 32 years) one has to show practical engineering relevance, which despite mathematicians stating the opposite, does hold for FLT (and maybe for Goldbach, which was so close to FLT that I couldn’t resist). – NB

PS: You’re welcome to try and apply my residue-and-carry method to Riemann. Let me know the result. In Holland we say (in Dutch of course): “you never know how a cow catches a hare”. In other words: keep it simple, you may be pleasantly surprised ;-)

Posted by: Nico Benschop on July 1, 2009 3:24 PM | Permalink | Reply to this
Read the post Elsevier Pays for Favorable Book Reviews
Weblog: The n-Category Café
Excerpt: Elsevier offered gift certificates to academics to write 5-star reviews of a book on Amazon and Barnes & Noble.
Tracked: July 2, 2009 11:48 AM

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Interesting John (assuming you put up this message about Elsevier): proof by proximity? If I were you, I’d be more careful with indirect accusations. Tsj, tsj, John: sloppy! It does not become a real mathematician (assuming you are one;-) Period. – NB

PS: would you care to read the proof first? (book chap.9). And while you are at it, also that of FLT (chap.8), and an extension of FST mod p to mod p^k for k>1(chap.7). With semigroups this becomes really simple undergrad stuff, and should not take much of your time. That would render you more qualified to make remarks, even constructive ones - although that may be too much asked from a real mathematician… (I do not mean removal of the diaeresis on the e in hiërarchy, which is of the usual nit-picking kind).
Notice that Springer has an online version of each chapter, so you don’t have to buy the whole book (heaven forbid: State-machine theory and planar Boolean logic ?-)

Posted by: Nico Benschop on July 2, 2009 1:50 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I hope people from outside mathematics don’t get the impression that Benschop’s not getting a fair hearing for his proofs. At least for the FLT proof, several mathematicians (Alf van der Poorten and Robin Chapman) have debunked Benschop’s work. At least as of a few years ago, when he first managed to get it into print (in Acta Mathematica Univ. Bratislava, which I’ve never heard of in any other context), the proof was not only wrong but showed no progress or even promise of future progress. I’d be shocked if the proof in his new book is any better, but I have no intention of paying \$25 for a 16-page chapter to find out.

In any case, the real issue here isn’t whether his proof is valid. Rather, it’s why a reputable scientific publisher would publish a technical book seemingly without any substantive review of its contents. (Acta Mathematica Univ. Bratislava has no reputation to protect, and may be run by crackpots for all I know, but Springer has a good reputation.) Books are often not refereed as carefully as journal articles, but there’s no excuse for publishing proofs of FLT and Goldbach that have not been endorsed by any mainstream mathematician whatsoever.

Posted by: Anonymous on July 2, 2009 3:21 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Hallo Anonymous,

You say “Acta Mathematica Univ. Bratislava has no reputation to protect, and may be run by crackpots for all I know.” Well, especially since you hide behind ‘Anonymous’, *you* may be a crackpot for all I know. What kind of language is this? I’ve seen enough denigrating stuff thrown at me the past 10 years, mainly from ‘mathematicians’ (sci.math and other forums) that I grew a hard skin. Regarding the U-Bratislava math department: they are the specialists on application of semigroup theory to elementary number theory, since their leader was prof. Stefan Schwarz, in the sixties the initiator of the journal ‘Semigroup Forum’.
You may not know it, of heard about it, but the very concept of ‘semigroup’ (=associative closure) comes from Eastern Europe: the first paper on the detailed structure of simple semigroups (viz. without proper ideal) was written 1928 by Schushkewitch in the Ukraïne. So in your own interest, stop blabbering senseless stuff and insulting people and institutes you don’t even know.

PS: I would like to ask the ‘moderator’ of this list (John Baez?-) to screen the comments placed here, at least on polite language, and maybe even on relevance, and certainly skip Anonymous entries.
– NB

Posted by: Nico Benschop on July 2, 2009 6:57 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Nico wrote:

PS: I would like to ask the ‘moderator’ of this list (John Baez?-) to screen the comments placed here, at least on polite language, and maybe even on relevance, and certainly skip Anonymous entries.

We always have plenty of anonymous and pseudonymous posters here; I don’t like it, but I don’t intend to ban it. On the other hand, direct insults of other people posting comments here is strictly forbidden. Saying that some department somewhere “might be run by crackpots for all I know” is acceptable, barely. Saying that someone you are talking to “may be a crackpot for all I know” brings you much closer to the point where I’ll delete a comment. And if I get tired of monitoring people’s behavior, I’ll just close down the thread entirely.

Posted by: John Baez on July 2, 2009 11:15 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I’m sorry for the implied insult to the University of Bratislava. It turns out that Acta Mathematica Univ. Bratislava, which I thought I’d never heard of, is the same as Acta Math. Univ. Comenianae, which is a journal I have indeed seen in the stacks in libraries.

This raises the question of what happened there, as well as at Springer. We don’t have free access to the book, but we do to the paper:

http://www.emis.de/journals/AMUC/_vol-74/_no_2/_benschop/benschop.html

It’s noteworthy how obscurely the proof of Fermat’s Last Theorem is announced in the paper. The only mention in the abstract is in the phrase “shown to imply the known FLT inequality for integers” on the 16th line of the abstract (which is a genuine mention but not terribly clear, since “the known FLT inequality” is not standard terminology and is not explained). There is no discussion in the introduction or further indication that a proof is forthcoming until Section 4, after 12 pages.

My guess is that the referee didn’t read the paper carefully and didn’t even realize it was claiming a proof of Fermat’s Last Theorem. Of course, I have no evidence for this, but it’s a plausible explanation. If the AMUC editors knew they were accepting a proof of FLT, why wouldn’t they publicize it (after all, it would be by far the greatest paper ever submitted there) or defend it (since the belief that they published a bogus proof hurts their journal)?

I haven’t looked at the Springer book, but the 2-page preface is available online for free. The only mention of Fermat or Goldbach in the preface is the following sentence:

“This balanced approach to arithmetic provides new insights into old and well known problems in finite additive number theory (Fermat, Goldbach, Waring: Chaps. 8, 9, 10) with practical engineering results.”

If you’re going to publish a proof of Fermat or Goldbach, I strongly recommend making it as clear as possible that you are doing so, so that nobody can imagine that it was accepted by accident (due to sloppy review).

Posted by: Anonymous on July 3, 2009 5:32 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Fair enough John, thanks. – NB

Posted by: Nico Benschop on July 4, 2009 9:36 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I’d like to give a brief plug for anonymous commenters considering pseudonymity instead. Pseudonyms are much more convenient for referring to other people in threads. It also allows you to build a reputation as someone worth listening to online. At the same time it protects your identity nearly as much as anonymity would.

Posted by: Noah Snyder on July 6, 2009 1:34 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The trouble is that unless you are willing and able to provide free access to the relevant material, people here may not be willing to spend 139 US dollars just for the purpose of trying to refute the claims.

Can you provide names of professional mathematicians who have examined your claims and are willing to vouch for their validity?

There is at least one very fine mathematician, Robin Chapman, who has examined claims you’ve made in the past (contained in various drafts of papers you’ve put on the arXiv and in sci.math postings) and considers them seriously wanting. Your book would seem to be about very similar material; given the professional standing of Dr. Chapman, some degree of skepticism doesn’t seem all that out of place.

But in the end, this post is not about you, it’s about the practices of Springer Verlag. A big question is whether Springer made any effort to ask mathematicians to review the dramatic, eyebrow-raising mathematical claims made in the book. Can you address this?

Posted by: Todd Trimble on July 2, 2009 3:28 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Nico wrote:

PS: would you care to read the proof first?

Sorry, I don’t have time. I didn’t claim your proofs are wrong — though I’m willing to bet they are, purely as a matter of gambling. And my criticism was not directed at you. It was directed at Springer Verlag, for not adequately refereeing your book.

How do I know it was not adequately refereed? Simple:

If your proofs are wrong, Springer should have caught this. Why? Not because publishers should catch every error, but because Goldbach’s Conjecture and Fermat’s Last Theorem are extremely famous. If your proofs are wrong, it will be very embarrassing for Springer to have published them!

On the other hand, if your proofs are correct, Springer should have announced this loudly to the whole world. News of your work should be on the front page of every newspaper! It would be a huge triumph for Springer.

So, either way, Springer did the wrong thing by letting proofs of these results appear quietly, without having a team of experts carefully check them and then formally announce to the world that an earth-shaking mathematical event has occurred.

How could they have made such a huge mistake? That’s the interesting question.

Posted by: John Baez on July 3, 2009 2:10 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I wouldn’t take quite that position, because Springer-Verlag is a general science and engineering publisher. It strikes me as conceivable that if everyone directly involved in reading the manuscript at S-V was in the Digital Communications department and none of their chosen referee’s commented on it, I can imagine the fact that it contained “groundbreaking” FLT and Goldbach results wouldn’t reach anyone with the mathematical background to see what a big issue new proofs would be. (The one guy I knew in VLSI design was very clever but who came in to computer science via (English-style) EE degree and it wouldn’t surprise me to learn that he doesn’t know the iconic status of the FLT and GC.) What I would say is that it seems unlikely that anyone with a background to understand in enough detail to properly review the FLT/GC sections ought to know, and if the chosen referee didn’t have the background to scrutinise those sections then they ought to have recused themselves on those sections.

Incidentally, this personal history reminds me of the time that a UK Professor of CS, who had been a physicist for many years before converting to CS, proudly regaled the pub with a subtly flawed version of the proof by contradiction of the infinity of primes, highlighting the difficulty in spotting the flaws in a “proof” even for very smart people.

Posted by: bane on July 3, 2009 3:40 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Hi bane, and John:

I don’t think you are right bane, since I did warn the editors specifically that not only a direct FLT proof was included as one of the chapters (ch.8 which does not mention FLT in the title because it in fact derives the full additive structure of the units group mod p^k, as *is* mentioned in the title - plus of course the extention to integers that is mentioned…), but also one for Goldbach’s Conjecture (ch.9). Now the FLT proof had already been published in 2005 by an established (!) math department (U-Bratislava), so that was no risk. Moreover FLT had already been settled by Wiles in 1995 (I did mention that just in case;-) The Goldbach proof of course was another matter, but since it used the same residue-and-carry method as applied in the FLT proof, I guess (but I do not know) they had it checked and found it worth a calculated risk to go ahead. Mind you, that proof is embedded (ch.9) in a whole lot of similar stuff on arithmetic, Boolean Logic and State-machines that form an associative algebra cluster, useful for computer engineers: the main purpose of the book. Fermat and Goldbach, using my associative function composition approach, are not at all difficult (otherwise I would definitely not have found it, in fact I just stumbled by chance over them;-) John and other professional mathemeticians suffer, in my view, from tunnel vision - the very reason they can’t imagine a shorter proof than Wiles’.
Function composition algebra (semigroups) just does not normally pop up in these corners of elementary number theory.
As the anecdote says: you may be looking in the dark of evening under a street lantern because there is the light (=known methods) but the ring you’re looking for may not be there at all! – NB

Posted by: Nico Benschop on July 4, 2009 10:06 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

What on earth do you mean by “additive structure” of the multiplicative group? It seems that the description you gave for the benefit of GS does not make any sense.

Posted by: EE on July 4, 2009 11:50 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Dear EE: I hope you agree that the group of units mod p^k (odd prime p, any positive integer k, called ‘precision’ over base p notation) is a subgroup of the p^k residues mod p^k that *also* have a well defined addition operation, namely normal integer addition with subsequent putting the produced carry (if nonzero) to zero. In the present case the additive structure of this field-subgroup G of units is characterized by the successor function S(n)=n+1. Taken together with the two symmetries: complement C(n)= -n and inverse I(n) = 1/n (both mod p^k) this yields eventually the ‘triplet’ structure (a+1)b == (b+1)c == (c+1)a == -1 with abc == 1 (mod p^k) for a,b,c in G.
This is due to the unique 3-cycle property of the above three *functions* S,C and I, of which IC == CI commute and S does not commute with C nor with I.
Then, upon complete inspection, each of the four possible compositions of all three: SCI, CIS, ISC, CSI has a period of 3 if repeated, e.g. (SCI)(SCI)(SCI)==E where E(n) == n is the identity function, and SCI = 1-(1/n), composing from left to right. I hope I’ve been clear enough. – NB

Posted by: Nico Benschop on July 6, 2009 10:01 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Dear Nico Benschop, I can appreciate that an electrical engineer is much better aware of the notion of carry. But it isn’t that mathematicians have totally ignored it. There is the theory of “Witt rings” which incorporates it in a significant way(cf. J.-P. Serre, Local fields).

Your claim of proof of the congruences modulo p^n are plausible. But there are difficulties in lifting what is happening mod p and p^n, to what is happening over Z. Sometimes it is not enough that something is true over each p-adic completion, for it to be true over Q(cf. Selmer curve). So one has to be careful. But it might as well be true that you did succeed to prove FLT in spite of all this; even then there remains the fact that some people have checked and refuted your proofs. Now to redeem your work, the best way is that somebody else of good reputation checks it and declares it to be of merit. Until then, the public must be skeptic.

For results like the Goldbach, or the Riemann hypothesis, there is always an asymptotic character about the methods employed so far; indeed in the equivalent form of the strong version of the prime number theorem, Riemann hypothesis is very much an asymptotic statement. The methods in a finite space usually fail for asymptotic results; such is my limited experience.

You, being an electrical engineer, obviously know quite a bit of mathematics and have spent some serious time on it. I wouldn’t want to discourage you from proceeding. However, I also find it strange that Springer published your proofs of FLT and Goldbach without verifying the veracity. Even if your proofs are wrong, the rest of your work in academia would hold you in good stead. But, Springer shouldn’t have taken it in without more verification. There are any number of crackpots “proving” such results, and moreover there are still circle-squarers and angle-trisectors still existing. Claims of proofs of such major theorems cannot be given a light ride.

I am also an outsider to this blog; so I should also point out that this may not agree with the mainstream view here.

Posted by: GS on July 2, 2009 8:03 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I should make clear that “the rest of your work in academia would hold you in good stead”, if it is the case that the work in question is of merit. However I should make it clear that since it is mostly in electrical engineering and related topics, I have nothing to say about it. That would be up to the experts to pronounce.

Posted by: GS on July 2, 2009 10:27 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Dear GS and Todd,

First regarding Todds question on: “A big question is whether Springer made any effort to ask mathematicians to review the dramatic, eyebrow-raising mathematical claims made in the book. Can you address this?” - Sure I can: I know as much of the reviewers Springer engaged as you do about those that read any paper you submitted to a journal, thus ZIP.

Next point: *why* do you want another authority than yourself to say it is OK? Think of Wiles’ >150 page proof (indirect via Tanyama-Shimura conjecture about elliptic curves and modular forms having a 1-1 mapping). It is said that only a handfull of specialists could verify it, and when the white smoke curled to heaven it was announced OK. So are *you* happy now. Q: why were you interested in FLT in the first place? Is it just like Mt.Everest (why climb it? Because it is there;-) Would not you rather verify a proof yourself, if it’s important to you?
For a Wiles type of proof: forget it.

Let me repeat a quote from Carver Mead (at CIT if I’m not mistaken), which I found so to-the-point that it is among the six quotes on the dedication page:
“It always worked out that when I understood something, it turned out to be simple. Take the connection between the quantum stuff and electrodynamics in my book. It took me thirty years to figure out, and in the end it was almost trivial. It’s so simple that any freshman could read it and understand it. But it was hard for me to get there, with all this historical junk in the way.”

I feel with him: important stuff you must be able to check yourself! I’m an EE with formal self-education in discrete math (associative algebra’s of various kinds, for my research in digital VLSI design methods). I’ll tell you something surprising: I did not look for a proof of FLT, I just stumbled over a property of three p-th powers mod p^k summing to zero (k-digit number representation base prime p) to find a better way of multiplying numbers than by a binary array-multiplier of n x n Full Adders, which is *much* to powerful for its purpose. A log-arithmetic scheme seemed useful, with a multiplicative generator, in math called a primitive root (see chap.7, I got a US-patent out of that within a year!). The group of units mod p^k (any k>0) is known to be cyclic (1 generator) and Fermat knew this! Its period equals (p-1)p^{k-1} because there are p^{k-1} multiples of p (hence not units). If 3 divides p-1 (p=1 mod 6) then there is a 3-cycle as subgroup, and any subgroup sums to zero (mod p^k). So there you have it: 3 p-th powers adding to 0 mod p^k (k>1), with the extra property that Exponent Distributes over a Sum (EDS property): (x+y)^p == x+y == x^p +y^p (mod p^k). That, in January 1994, reminded me of Fermat, and I had to make a decision whether I would spend company- (and private-) time to “go for FLT” or not.
With the EDS property it should be possible to show the impossibility of extending such cubic root of 1 mod p^k type solution to integers. It took a while but you can, without high-math verify it for yourself (both cases 1 and 2 of FLT).

These are solutions for p=1 mod 6, but there are also another type of (more general) solutions for some primes -1 mod 6, starting with p=59 (more complex, but cubic roots are a special case), of which I’m sure Fermat had no clue. Possibly he realized that to present his cubic-root proof he would have to show they are the *only* solutions, which he could not… so he left it at rest (not divulging what he knew already). For p=7 and arithmetic mod 7^2 the first case of FLT case1 occurs, easily done and found by hand.

Now I was then 55 years old, a senior researcher (having parttime tought 6 years at Delft University) so I thought: I’ll give it a half year upto a year (between other research). And by december that year I submitted my paper to the internal screening board, with a math prof. of Eindhoven Univ. as advisor. He immediately blocked it, even though I showed the engineering relevance for a log-arithmetic scheme based on (p+1^)* generating a cycle of period p^{k-1}, namely all k-digit numbers that are 1 mod p, implying that each odd binary coded number equals a unique power of 3 (mod p^k). He said: if *that* were true we would have known this years ago (the proof is half a page!) So then my ‘paper polishing’ phase began, which lasted some 10 years… until publication by the experts in semigroups applied to elementary arithmetic: Univ. of Bratislava! (re: prof Stefan Schwarz).

The clue for Goldbach came also by chance: I had only to change the mudulus to \prod(first k primes) and first solve Goldbach-for-residues (mod m_k): each even residue equals the sum of two units,
whic took me just a few weeks, and another polishing phase started,in 1996,
until last year Springer (first Germany, then Holland: Kluwer Academic publishers before 2004) accepted in august my manuscript. So if you’re interested: chapter 9 has all the introductory semigroup / lattice theory you need for the proof: no other books necessary.

It was nice talking to you both as civilized people do. – NB

Posted by: Nico Benschop on July 2, 2009 11:40 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Nico, thanks for the reply. I asked in the first part of my comment whether you could name one professional mathematician who has examined your claim of having proved FLT and Goldbach, and is willing to vouch for its validity. This question you didn’t answer (you don’t have to, of course).

The question I asked at the end (do you know whether any professional mathematicians reviewed your book) was not asking whether you knew the identity of such reviewers, just whether there were such reviewers. (I fully expect the answer is ‘no’; the question actually verges on being rhetorical.) If I submit a paper for publication, then if things go as they should, I will in time receive a referee’s report. I don’t know who the referee is, but at least I can tell from the report whether my paper was reviewed. I was asking you whether there was any indication that someone, never mind who, examined your claims about FLT, etc.

Again, my guess is that no professional mathematician was consulted. This, while I’m sure it’s true, is quite amazing to me. The relevant staff at Springer would have to know the notorious status of FLT, and it just seems scandalous in the extreme just to let slide the remarkable claim that after hundreds of years of failed attempts to discover an elementary proof of FLT, an electrical engineer has found one. It could be true – I can’t rule out the possibility – but I think it has to be admitted that it would be remarkable indeed. Such a claim should undergo careful scrutiny, and I fully expect they completely dropped the ball, or ignored it, or whatever. Which would be just incredible and completely shameful behavior. This has nothing to do with you particularly – I’m commenting here on the actions undertaken (or not) by Springer Verlag.

A full response to other parts of your letter would get into the sociology of mathematics, which I’m not sure I want to get into that much; I’ll try to be brief.

Next point: *why* do you want another authority than yourself to say it is OK? Think of Wiles’ >150 page proof (indirect via Tanyama-Shimura conjecture about elliptic curves and modular forms having a 1-1 mapping). It is said that only a handfull of specialists could verify it, and when the white smoke curled to heaven it was announced OK. So are *you* happy now. Q: why were you interested in FLT in the first place? Is it just like Mt.Everest (why climb it? Because it is there;-) Would not you rather verify a proof yourself, if it’s important to you?

For a Wiles type of proof: forget it.

First, I never said I was interested in FLT in the first place, or that it was important to me. I certainly don’t have a professional interest in the problem. I have what you might call a general interest, in the sense of enjoying being a spectator of what great times we live in, when problems such as FLT or the Poincaré conjecture are at last surmounted, and I also have some interest in one day acquiring an idea of the proofs at least in outline. But that doesn’t mean I have a burning interest in working through the details of papers which presuppose a background I don’t have. (There are other parts of mathematics where I have acquired some substantial training, and it’s on the outskirts of that where I work and spend my time. Life is short after all.) So yes, I’m perfectly willing to take it on faith from the real experts whether a substantial claim has been validated, in an area where I have no expertise.

And here too, again because life is short, I’m perfectly willing to accept expert testimony from Chapman and van der Poorten regarding your own proofs. I am certainly not willing to spend time and money trying to work through a purported proof that I have absolutely no faith in to begin with – for various reasons I have no faith whatsoever that an elementary proof of only a few pages exists for FLT or Goldbach. If a number theorist does come along and vouches for your proof, then at that point I very well might take a look myself, but as I say I have no professional interest in the problem, and I choose to spend my time pursuing something else.

You have been a professional yourself (in electrical engineering, I understand), so I’m sure you can appreciate this.

Posted by: Todd Trimble on July 3, 2009 2:52 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The relevant staff at Springer would have to know the notorious status of FLT, and it just seems scandalous in the extreme just to let slide the remarkable claim that after hundreds of years of failed attempts to discover an elementary proof of FLT, an electrical engineer has found one. It could be true – I can’t rule out the possibility – but I think it has to be admitted that it would be remarkable indeed.

If nothing else, they should have informed the marketing department!

Posted by: Toby Bartels on July 3, 2009 5:16 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

FWIW, I agree with what Todd says. By my own very subjective estimate, I would guess that about 99% of all music, movies, literature, etc. ever made is probably not worth my own time because life is short.

Does anyone have even a rough sense from the history of math or science what proportion of “claims” stand the test of time?

For example, I recently asked an expert in both category theory and the math of string theory a new question which he said is “very interesting”, but there are three immediate problems with this question: neither he nor I can yet see how to answer my question in a concrete way, there is no experimental evidence for string theory, and there is no compelling evidence that category theory must be necessary for physics.

Posted by: Charlie Stromeyer on July 3, 2009 1:59 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Nico wrote “Next point: why do you want another authority than yourself to say it is OK?”

This is something that people tend to get the wrong idea on. It’s not that it’s an authority “as an authority”. Proofs are by their nature very difficult to pronounce a considered view on their correctness because finding flaws in a proof requires going against the flow of the writers argument and spotting problems with issues that may not even by explicitly mentioned in the proof (a single “step” may tacitly involve lots of assumptions). If I sat down with a textbook I could probably spot a glaring problem but I’d say the odds are low-ish that I’d be able to spot a subtle problem, and for a very subtle problem I’m prepared to say I might very well miss something even in my area of expertise. (I’ve read a couple of papers in my area that provide “proofs” of existing theorems a new way that I believe contain (trivial) flaws that got overlooked partly because they knew they were re-proving a true result, so alternative proofs of known true statements are especially likely to be incorrect.) So knowing that at least one person with intimate knowledge of the issues has spent a long time challenging the proof and not found any problems is important to know.

The idea that you can really just verify everything yourself without external reference doesn’t work beyond relatively simple arguments.

Posted by: bane on July 3, 2009 10:17 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Dear Benschop,

Wiles’ proof of FLT is grandiose because it leads to a proof of the Taniyama-Shimura conjecture, which is great. It is extra nice that Taniyama-Shimura imples FLT; but even otherwise this conjecture is important. The achievement of Wiles will stand, even if there is an elementary proof.

And, I do know a little bit of elliptic curves; so I do understand some of the general philosophy of the proof, and find myself thinking of the many connections from time to time. So much I can verify with my own thought, and the rest I can trust the judgment of the scores of established people working on that subject. So I can convince myself that Wiles’ proof is probably correct. Also I intend(slowly) to understand more of the subject.

However, in your work, I failed to see any connections of astonishing and surprising depth, which one sees in the work of Wiles, and the people who laid the groundwork for him(“on the shoulders of giants..”). One has difficulty in believing that the profound methods can be replaced by half a page of computational considerations.

I must confess that I do not understand your talk of various cases of FLT, references to Chapters etc.. Would you be able to give a source where I can read it without paying money?

Posted by: GS on July 2, 2009 11:56 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Dear GS, you write “Would you be able to give a source where I can read it without paying money?” – Sure, the moderator of sci.math.research gave a link to my Goldbach paper in arxiv.org :
http://arxiv.org/abs/math/0103091 where you find an older version (2001), and links to various intros on my homepage. That is an early version, only in non-essential details differing from the present book version. If you search for ‘Benschop’ there, another 7 papers appear, including that for FLT. BTW: I derive much more than FLT, namely the full additive structure of the group of units mod p^k, characterised by a threefold successor function I call ‘triplet’: (a+1)b == (b+1)c == (c+1)a == -1 with abc==1 mod p^k. Of which the cubic roots form a special case, when a==b==c follows: a+1=-1/a (with 1/a == a^2). If you have any questions, please let me know. – NB

Posted by: Nico Benschop on July 3, 2009 7:44 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

To GS. You say “However, in your work, I failed to see any connections of astonishing and surprising depth, which one sees in the work of Wiles, and the people who laid the groundwork for him (… on the shoulders of giants…;). One has difficulty in believing that the profound methods can be replaced by half a page of computational considerations.”

NB: It’s not ‘half a page’ but 16 pages, yet including the full additive structure of the units group in Z(.) mod p^k.
The key in my work is to treat the two symmeteries of residue arithmetic as *functions* : complement C(n) = -n, and inverse I(n) = 1/n, together with the successor function S(n) = n+1.
Then by inspection you can verify that I and C commute: IC = CI, leaving four possible combinations of these three functions, namely SCI, ICS, ISC, CSI which *all* have a period of three :
denote the identity function E(n) = n, then for instance (SCI)(SCI)(SCI)=E,
where working left-to-right: SCI = 1-(1/n), repeatedly applied to itself (substitute n by 1-(1/n) three times). From this treatment of the two symmetries of residue arithmetic I derive the ‘triplet’ structure (see my other msg today) as the only possible additive structure of the units group. The integers come in by carry extension *and* the EDS property of such additive solution (including FLT mod p^k type solution, which *must* hold also for any integer solution) makes preservation of equivalence for large enough modulus mod p^{pk} impossible. Notice the p-th power of a k-digit number x smaller than p^k requires at most pk digits (base p).

And regarding a complex problem reduced to a much simpler one: think of the 77 circle/epicircle model of Ptolemeus for the orbits of the known five planets. Just by making the basic circle into an ellipse Kepler needed only one elliptic orbit per planet, enabeling Newton to reduce that further to a simple inverse quadratic law of dynamic motion (his gravity law). In this FLT case: arithmetic is associative and commutative, and it requires the one level more powerful algebra of function composition, which is not commutative, to solve hard additive number theory problems
(here: S does not commute with I nor with C, where successor function S is characteristic for addition). In other words: “You need steel to cut wood, and you need a diamnant to cut steel” (where Boolean logic = wood, arithmetic = steel, and diamant = function-composition (semigroups). – NB

Posted by: Nico Benschop on July 3, 2009 10:47 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I do not understand what you mean by Hensel extension lemma preventing solutions of FLT. In any case, if what you mean is the following lemma:

then you can see that the condition for applying this lemma is not satisfied, because the derivative mod p of the Femat equation vanishes, whatever the variable you use for taking the derivative. So it seems that you cannot use this lemma here.

Posted by: GS on July 3, 2009 10:30 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

You refer to, as it says, * one version* of the Hensel lemma. The simple version that I refer to (Hensel’s extension lemma, or shortly: Hensel-lift) does not require a derivative. It says that if an equivalence F(n) == G(n) mod p^2 (prime p) holds, then this can be extended to equivalence mod p^t for any t>2. For his p-adic number theory Hensel extends to infinite precision (inf t).
This has been a block to looking at carry extensions to (finite) integers, because any precision t can be made to hold. However, in the case of an equivalence (x+y)^p == x+y == x^p + y^p mod p^k (k>1) where the exponent distributes over a sum (EDS property), taking the p-th powers of a k-digit residue solution yields already inequivalence mod p^{3k+1}, which you could call a ‘bootstrap’ effect: for any prcision k of a FLT mod p^k solution, its p-th powers (taken as integers) cannot preserve the quivalence beyond triple precision 3k+1.

And re the ‘structure’ of the units group G of mod p^k, I mean its *additive* structure, not its multiplicative structure, which is cyclic of order (p-1)p^{k-1. Hence G is the direct product of two cyclic components G = A.B where A is the ‘core’ of order p-1 with n^p == n for all p-1 in core A (FST extension to mod p^k), while the ‘extension component’ B = (p+1)* is generated by p+1, yielding p^{k-1} distinct powers of p+1. So if A == g* (primitive root of core A) then each unit residue n = g^i.(p+1)^j for unique pair of naturals i \lt p-1, j \lt p^{k-1}. But that is its * multiplicative* structure, not the additive one. – NB

Posted by: Nico Benschop on July 3, 2009 11:22 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Also I should point out that the structure of the group of units mod p^k is known. Again, I do not understand the description you give; but the structure is simple enough, and can be found in any algebraic number theory book. I hazard the guess that your description, once unravelled, would not agree with the actual one.

Posted by: GS on July 3, 2009 10:38 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The proof I know of Hensel’s lemma uses Taylor approximation, and the nonvanishing of derivative is crucial therein. If you make a stronger claim, you have to prove it.

But, even assuming it, Hensel’s lemma as stated is for polynomials in one variable. How do you use it for the Fermat equation, given in three variables?

I suppose if I am to read the whole paper like this and keep querying you, the blog readers might not like it. So, to be brief, I see that somebody else apparently went completely through your proof and found errors(Robin Chapman?). Were you able to refute his objections?

Posted by: GS on July 3, 2009 1:22 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

These discussions with Robin Chapman took place some 10 years ago on the sci.math newsgroup. He did indeed find errors, mainly due to misunderstanding my intent, which was not expressed accuratly enough. Such errors were readily corrected.

For instance (Ch.7 on the extension of FST mod p to mod p^3, removed from Ch.10 that was published, and where this was a sideline subject): in search of primitive roots of 1 mod p I considered divisors of p-1 resp. p+1, since these are powerfull generators in the units group G mod p. This I thought to compress into: divisors of p^2-1, which is in error because such divisor can contain prime divisors of p-1 and p+1 about which product I had proven nothing. So p-1 and p+1 should be considered separately. Trivial, and easily corrected, but essential. A second point relating to this was my claim that different divisors r of p-1 would have distinct r^{p-1} mod p^3. In fact this should be reduced to: divisors with different prime-divisor sets, hence generating distinct idempotents in Z(.) mod p-1.

PS: the unpleasent memories I have from those discussions was his denigrating tone towards me (as most mathematicians still do): the total lack of civilized discussion style. But I got used to that, and thanked him for his efforts. – NB

Posted by: Nico Benschop on July 5, 2009 11:29 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Hmm. I am unable to check more of your paper myself, as I am finding myself confused by your terminology, which is a bit unfamiliar to me. Therefore I am not able to say more. But I must add that I am still skeptic about your results; Goldback or FLT didn’t get proved so easily for centuries.

Posted by: GS on July 6, 2009 4:27 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

“Goldback or FLT didn’t get proved so easily for centuries.” (it’s Goldbach;-)

The problem, I think, is that mathematicians know too much to appreciate a (relatively) simple approach. For instance your and my ref to the Hensel-lift are totally different. For me suffices, as extension lemma: F(n) == 0 mod p^2 implies an infinite number of solutions mod p^t for all t>2. Nothing more and nothing less, while your ref is almost infinitely more complex, with derivatives and Taylor series and what not. You make it much to dificult for yourselves. Does one not say: Truth is simple (“In der beschränkung zeigt sich der Meister” - Göthe, I believe).

The reason that FLT and Goldbach were not solved yet is because people made it themselves much too difficult (look at Wiles’ proof: that is extremely indirect, going to the next room via the Northpole !-)

More powerfull tools than arithmetic (associative and commutative) are needed to crack these additive number theory problems, which ‘naturally’ points to just associative (= function composition = semigroups), and take the successor function S(n)=n+1 as basis for additive analysis, as related to the two symmetries: complement C(n)= -n and inverse I(n)=1/n. This tool is already more than 100 years available, just USE it!!! – NB

PS: I bet that a system with m symmetries that distribute in sequential (cascaded) order over each other, would yield an m+1 cyclic additive structure, if the bottom operation is (+) with complement as symmetry, just as here with 2 symmetries a 3-cyclic ‘triplet’ additive structure follows: (a+1)b == (b+1)c == (c+1)a == -1 with abc == 1 (mod p^k) for prime p. – NB

Posted by: Nico Benschop on July 6, 2009 5:03 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

It is true that Wiles’ proof leads much farther away than the question at hand. It would be very desirable to have a more direct proof. And you claim to have one. However, what can I say to you if I do not understand it? I see a few statements which are correct; but the rest of it is gibberish for me. It may or may not make sense; and I suspect that you are in error somewhere.

You are an electrical engineer, and you have published a proof of two major theorems, FLT and Goldbach, motivated by considerations of digital systems design. Imagine the following scenario, which is extending yours. Suppose one day a publication of some lawyer appears, claiming to solve all the ten Clay problems. And he claims to be motivated by the method of “jurisprudent fluxions and tort reciprocity”. And each problem has 14 or so pages devoted to it. And he claims, mathematicians make things too complicated; that is why they cannot understand or appreciate or this work, and this is precisely why they couldn’t prove it earlier. And suppose Springer publishes it. I do not know whether imagining this scenario will succeed in shaking your conviction, or will it merely strengthen it, as had been happening all along?

Sorry about the denigration you have suffered at the hands of mathematicians. It is true that sometimes genius comes out of nowhere. But, there is the following factor. When some unknown genius contacts a professor with proofs of so-and-so theorems, it is usually the case that he has many results to show. His terminology and style might be strange, and he may claim to prove a major theorem. But, if he is such a good mathematician, isn’t it natural to expect that he is able to prove some other unknown results as well? The same methods should prove some not-so-hard, but nontrivial results. Also the methods should be clear for the onlooker. As far as I can see, you haven’t proved any arithmetical theorems other than FTL and Goldbach. Claiming to prove a major result out of the blue is what cranks do. Of course, there is a probability that you are right, but it is small; especially so when you are a person who proves only big results.

Posted by: GS on July 6, 2009 8:49 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

GS: …”But, if he is such a good mathematician, isn’t it natural to expect that he is able to prove some other unknown results as well? The same methods should prove some not-so-hard, but nontrivial results.”

OK, after getting grip on FLT, I thought indeed: I don’t need the Goldbach result for my research, but let me see if the method suffices also for Goldbach: by just adapting the modulus for the residue part of the proof: each even residue is the sum of two units mod m_k, where m_k = \prod(first k primes). That worked indeed, as did the subsequent carry extension (called ‘prime seive’ in the manuscript). What more do you want, is that not enough, GS?

If you look at other chapters you’ll see more unsuspected new results, for instance in the part on State machine decomposition, where folklore has it (Krohn/Rhodes 1965) that a permutation machine with a non-trivial simple group G as closure cannot be decomposed any further, hence is a primitive component. Now you probably know that the smallest such group has order 60 (namely A5, the alternating group over 5 objects). The reason is that their theory depends on group theory, where a full congruence is required for cascade decomposition. However, I show that only a right congruence (viewed as preserved state-partition, re Hartmanis/Stearns 1970) suffices, allowing decomposition of such permutation machines in a cascade coupled network of cyclic group machines (for A5: with period 2 [twice], 3 and 5). After all, A5 does have plenty of subgroups, and as many right congruences, so plenty decomposition possibilities.

Or see ch.5 where Boolean logic can be based on a square grid of switches (MOS transistors) in the plane *without* crossing wires (that would be expensive in planar silicon technology). I derive there that any Boolean function (BF) of at most 4 inputs can be made planar by inverting and/or permuting inputs resp. complementing the output. As far as I know the concept of planar BF is new and useful. Moreover, this BF representation allows solving BF problems in polynomial time (polynomial in the number of inputs), and I made a synthesis program to show that in practice. Interesting for P vs. NP ? – NB

Posted by: Nico Benschop on July 6, 2009 10:57 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

In any case, the additional result you claim is not enough: it is easy to prove/disprove the veracity by existing elementary methods. But, your claim about P = NP is interesting; you are making good progress towards all the Clay math problems, in the manner of our imaginary lawyer friend I mentioned to you.

In any case, it is interesting that Springer published it. The author is overzealous in defending his work prematurely(i. e., before achieving P = NP too and other results of similar repute). As regards Springer, it seems that in addition to being totally irresponsible about publishing such a thing, they also do not care a hoot about open criticism in a public forum. Good going.

Posted by: GS on July 6, 2009 11:20 PM | Permalink | Reply to this

### Krohn, Rhodes, and Minsky wrong? Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Ouch. This is getting too close for my comfort. Even if this is all Ad Hominem, I can’t keep silent. “Krohn/Rhodes” is denied validity? My PhD dissertation work, and several published papers, depend on Krohn-Rhodes decomposition of semigroups of differential operators. Marvin Minsky praised my work, and he was John Rhodes’s thesis advisor. So Krohn and Rhodes are wrong, I’m wrong, and Marvin Minsky is wrong? Suddenly I’m in the Establishment, and not looking forward to the alleged Revolution.

Posted by: Jonathan Vos Post on July 7, 2009 1:15 AM | Permalink | Reply to this

### Re: Krohn, Rhodes, and Minsky wrong? Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Well, the problem is that I was simply not aware of the Krohn/Rhodes work. Like much else of Benschop´s terminology.

Apologies, and I must withdraw from discussions of things that I am not knowledgable in. I hope somebody with knowledge of all what Benschop says will look into his claims.

Posted by: GS on July 7, 2009 7:41 AM | Permalink | Reply to this

### Re: Krohn, Rhodes, and Minsky wrong? Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

I took the time to look at Marvin Minsky. It seems that he is in artificial intelligence. The reference to computer science/electrical engineering situations too often isn’t going to help the situation if you want number theorists to understand the work.

Apologies to John Baez for spoiling his thread with a discussion with the author of the paper, and ruining the objective of Springer-bashing.

Posted by: GS on July 7, 2009 8:15 AM | Permalink | Reply to this

### Re: Krohn, Rhodes, and Minsky wrong? Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Minsky is (was?) primarily in artificial intelligence, but he has published some reasonably pure stuff on finite automata and learning algorithms. He is a perfectly reasonable authority to cite on the subject of finite automata.

Anyways, Krohn-Rhodes automata theory definitely seems to be a real subject. It brings up 99 hits on mathscinet, mostly in computer science journals, but also in pure math papers like this one http://www.ams.org/mathscinet-getitem?mr=1837243 .

Posted by: David Speyer on July 7, 2009 8:11 PM | Permalink | Reply to this

### Re: Krohn, Rhodes, and Minsky wrong? Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Jonathan wrote: “Krohn/Rhodes” is denied validity?”

NB: No that is not what I wrote. That decomposition theory of automata yields a network of permutation- and reset- machines, which are only two of the five ‘basic’ (indecomposable) components if you employ the full possibilities of semigroups. These basic state machines have as closure one of the five non-isomorphic semigroups of order 2, with perfectly applicable engineering functions: two counters (periodic and monotone), one pair of ordered idempotents (re AND cq. OR), and two non-commutative memory types: branch (if-then-else if binary) and reset (D-flipflop), storing the first resp. last input. The 2 elements can readily be extended for each type into n \gt 2: the five types of basic state-machines to build digital networks. The algebra used is that of associative function composition (=finite semigroups), hence the title of the book.

A corresponding coupling mechanism is different than the wreath-product, namely based on a mapping from the leading component into the automorphisms of the depending (trailing) component. Note that these principles are different from those of Krohn/Rhodes: which seem to preserve the state concept, which I ‘integrate’ into the full closure to functions as stateset (possibly extended by a left-identity). In fact the given stateset is a ‘defining’ right-congruence of its closure. This all is in chapters 1-4, the state-machine part of the book. – NB

Posted by: Nico Benschop on July 8, 2009 10:29 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

When some unknown genius contacts a professor with proofs of so-and-so theorems, it is usually the case that he has many results to show. His terminology and style might be strange, and he may claim to prove a major theorem.

My suspicion is that folklore here is misleading things. From the few cases that I know about, genuine geniuses from outside the established community (as opposed to those like Perelman who were conventionally trained and commnunity members, but just not high profile or gregarious ones) don’t turn up with proofs of existing conjectures but rather previously unknown mathematical theory (which may turn out to be different, unorthodox ways of writing known mathematics). Eg, my understanding of the letters referring Ramanjuan to Hardy only contained independent derivations of known results as well as many bizarre looking new results, but no claims of solving already known problems. I might hazard a guess that this is because it’s much, much, much more probable to come upon some way of doing things that completely hasn’t been spotted before than to spot some simple “missed links” (that are actually correct) within an already studied subject.

In fairness, I suppose number theory may be different in that many problem statements are understandable with relatively little reading of the established literature whilst the concepts and techniques used in proofs in the field are much, much more difficult. In contrast in many other fields the statement of the big problems actually requires a significant amount of work to understand it (eg, look over the Clay institute problems).

Posted by: bane on July 7, 2009 10:24 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

GS writes:

Of course, there is a probability that you are right, but it is small; especially so when you are a person who proves only big results.

I completely agree, but through long and sometimes painful experience I’ve discovered that when somebody claims to have proved two such big results, they rarely reconsider after hearing comments such as these. So, after a few more rounds of discussion I may turn off comments on this thread.

Posted by: John Baez on July 6, 2009 10:32 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Thank you John. I’m getting a bit bored too;-) Rather then explaining for the m-th time, if one is interested, just buy the book. And if that is not worth it, too bad for you. – NB

Posted by: Nico Benschop on July 6, 2009 11:04 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Before the thread gets turned off, I’d like to have a go at this. I believe Nico said that the proof of Goldbach’s conjecture in his book is not different in essence from a proof that appears in this paper. I am confused by some things in the paper, and imagine that I’d be confused by the same things in the book treatment. But since we have Nico here, I thought maybe I’d see if he could address these points.

Okay. I’ve tried to read this paper as sympathetically as I can, and in fact I understand more or less everything that appears in sections 1 through 6. In fact, there happens to be plenty of meaningful and correct content in these sections, although the reader may have to look past some rather unconventional use of language to get at it. (Of the many instances of unconventional language, I’ll mention just one: calling certain subsets $G \subseteq H$ of a semigroup a subgroup, when what is meant is that $G$ is a subsemigroup which happens to be a group, with an identity element which is not the same as the identity of $H$.) Probably Nico’s use of language has made it hard for many mathematicians to see what he is trying to say, with the result that they may be dismissive even when he is driving at something which turns out to be correct. I tell myself to remember that Nico is not coming from this culture and may not be fully aware of how we mathematicians use language. (However, if he wants people to listen, he should learn!)

To continue: after a little deciphering, I think a lot of what transpires in sections 1 through 6 can be translated into precise statements and correct proofs. Much of it can be seen on the basis of routine applications of the Chinese remainder theorem. For example, theorem 6.1 (“Goldbach for Residues”) is certainly correct. It says that for $m$ an even squarefree integer, $m = p_1 \ldots p_k$ where the $p_i$ are distinct primes and $p_1 = 2$, every even element $x$ in $\mathbb{Z}_m$ is a sum of two units in $\mathbb{Z}_m$. This is not at all hard to prove, using the Chinese remainder theorem, in the form that says that the ring homomorphism

$\phi: \mathbb{Z}_m \to \mathbb{Z}_{p_1} \times \ldots \times \mathbb{Z}_{p_k},$

defined componentwise by reduction mod $p_i$, is an isomorphism.

Things started becoming a lot harder for me to decipher in sections 7 and 8. In particular, I’m having a lot of trouble making heads or tails out of the purported proof of theorem 8.1. Here is the proof in its entirety:

Proof. Euclidean primesieve (4) is used, with induction over $k$ and induction base $k = 3$ (lem7.1). The induction step for any $k \gt 3$ restricts summands to units $\lt p_{k+1}^2$, the minimal composite in $G(k)$. So these units are all prime, denoted by primeset $P′(k)$ of all primes between $p_k$ and $p_{k+1}^2$.

Failure of GC [Golbach’s conjecture] for some $2n$ in $S′(k) = P′(k) + P′(k)$ for all relevant $k$ [italics mine; see below], so $p_{k+1} \leq n \lt p_{k+1}^2$, causes those $S_0(k)$ containing $2n \in S′(k)$ to be incomplete (lem7.2), along with all extensions $S_0(k) + 2c m_k$, yielding an incomplete $S_0(k+1)$. So, due to the Euclidean prime sieve structure (4) with $G_1(k + 1) \equiv G_1(k) mod p_k$ based on carry extension, the missing $2n$ will not be covered by residue pairsums mod $m_{i\gt k}$ either, contradicting thm6.1 (GR [Goldbach for Residues] mod $m_k$). Bertrand’s Postulate implies the $S′(k)$ for successive $k$ to overlap, establishing Goldbach’s Conjecture (GC). $\Box$

I’m having trouble following any of this, but let me focus on the part that I italicized. What does it mean?? It sounds like it should start off saying, “suppose Goldbach’s conjecture fails, i.e., that some even integer $2n$ is not a sum of two primes.” But then I see in the actual text “some $2n$ in $S'(k) = P'(k) + P'(k)$” which (translated) says that $2n$ belongs to $S'(k)$, i.e., is a sum of two elements belonging to $P'(k)$, which was defined in the preceding paragraph to be a set of primes. So the sentence in the text seems to translate to, “Failure of GC for some (even integer) $2n$ which is a sum of two primes belonging to $P'(k)$ for all relevant $k$…” Huh?? How does GC fail if $2n$ is a sum of… ?

Maybe it’s meant to say: suppose that $2n$ does not belong to $P'(k) + P'(k)$ for all relevant $k$; we will derive a contradiction? (What does it mean to say that $k$ is ‘relevant’?)

The italicized segment reads not as a supposition or an assertion, but as a noun phrase. How can something (“so $p_{k+1} \leq n \lt p_{k+1}^2$”) – how can anything at all – be a consequence of a noun? It’s not even grammatical. Please tell me: what is the actual assertion or hypothesis from which these inequalities follow?

I hope Nico will provide a clearer reformulation. Please, Nico, do not send me to your 139 dollar book; I can’t ask a book questions. Part of the process of persuading the mathematics community is to be able to personally address any questions that come up; now I am making a good faith effort to read your paper, and I hope you make a good faith effort to answer questions I may have.

Posted by: Todd Trimble on July 7, 2009 3:44 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Well Todd, you have a point there, but “2n in incomplete S_0(k)” makes slightly more sense in the booktext since the carry ‘c’ of the primeseive (9.4) is mentioned to specify better the range of integers involved: “Now assume GC to fail for some 2n in S’(k)=P’(k)+P’(k), hence S_0(k) containing 2n is incomplete (Lemma 9.6), along with all extensions S_0(k)+2c.m_k, yielding incomplete S_0(k+1). By primeseive (9.4) with G_1(k)+1) == G_1(k) mod m_k based on carry extension, the missing 2n would not be covered by pair sums mod m_i (i \gt k) either. But this contradicts Theorem 9.1 (Goldbach-for-Residues GR), establishing Goldbach’s Conjecture (GC).”

The context of the primeseive (9.4) is essential, and the phrase “some 2n in S’(k)=P’(k)+P’(k) ” must look like a contradiction, but is meant to indicate the integer range involved (‘relevant k’), re: “along with all extensions S_0(k)+2c.m_k, yielding incomplete S_0(k+1).”

Although improvement, in order to prevent misunderstandings, could be called for, the prime-seive context (9.4) should suffice to correct such. Those are the dangers of quoting out of context… – NB

BTW: I too was surprised about the high price of the book (for hard cover). Other books appear to have a soft-cover version of half the price, which my book does not have. So I pleaded the editor to consider this here too. The ability to buy separate chapters online is totally against my intention (it’s like selling a company in pieces to gain a profit;-), since I hope to reach interested people from various disciplines, especially students, to see the connection between the different fields (EE, CS, NT). In fact I pleaded with Springer to remove that online feature in this case, provided a cheaper soft-cover version would become available.

Posted by: Nico Benschop on July 7, 2009 9:49 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The text of Thm 9.2 (goldbach Conjecture) can be readily improved by replacing (in fact reducing) the first line of the last paragraph as follows:

“Now assume GC to fail for some 2n, hence S_0(k) is incomplete (Lemma 9.6) for some k, along with all extensions S_0(k)+2c.m_k, yielding incomplete S_0(k+1).”
I notice I made it too difficult for myself: simpler is better;-) – NB

Posted by: Nico Benschop on July 7, 2009 10:05 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Although improvement, in order to prevent misunderstandings, could be called for, the prime-seive context (9.4) should suffice to correct such. Those are the dangers of quoting out of context…

Let me add that it’s not good form to blame the reader (“quoting out of context”, I did no such thing) for failing to follow text such as this. Have some mercy for the reader. If you really care about your results, you will spare no effort to make them as clear and error-free as you possibly can – right?

Anyway, from your latest response, I guess the amended text is supposed to say something like this, “Suppose GC fails, i.e., that $2n$ is not the sum of two primes.” Followed by, “Then for some $k$, $S_0(k)$ and all of its extensions $S_0(k) + 2c m_k$ are incomplete [i.e., do not contain $2n$], and it follows from lemma 7.2 that $S_0(k+1)$ is also incomplete.” Do I have it right so far?

Okay, then we have

So, due to the Euclidean prime sieve structure (4) with $G_1(k + 1) \equiv G_1(k) mod p^k$ based on carry extension, the missing $2n$ will not be covered by residue pairsums mod $m_{i \gt k}$ either, contradicting thm6.1 (GR mod $m_k$).

I’m still not really following this, but I assume that $p^k$ is a typo, and that you mean $p_k$. Also, since you said you have arrived at a contradiction, why wouldn’t the proof end right there? In other words, after arriving at a contradiction, you could just say, “Therefore every $2n$ must be the sum of two primes, QED.”

Posted by: Todd Trimble on July 7, 2009 12:34 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Thanks Todd for your efforts. You are perfectly right that I should do everything to make unclear passages clear. So that’s what I am trying to do now. The point that must be stressed more is that, if S_0(k) is not complete (re lemma 9.6 about unit pairsums covering E(k) the set of even residues mod p exp k
– the power sign seems not printable here) due to no prime-pair available, such missing 2n cannot be implemented by another composite pair or prime-and- composite pair, since they are too large, namely beyond the smallest composite p_{k+1} exp 2. Only units between p_{k+1} and its square are used, since they are guarenteed prime, as the proof starts mentioning. I will have to find a compact phrase to express this restriction. – NB

Posted by: Nico Benschop on July 7, 2009 4:55 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Hi Nico – just a few things.

(1) If you can use only references internal to your paper (that I linked to in a previous message), that would be great. I don’t have your book, and don’t intend to buy it (sorry), so your lemma 9.6 is not visible to me, and I guess I’m mystified because I see nothing in the paper that corresponds to “re lemma 9.6 about unit pairsums covering $E(k)$ the set of even residues mod $p^k$”. (What is “$p$” here?)

(2) You seem to know LaTeX, and you can easily use it here by changing your text filter, say to “itex with MathML with parbreaks”. Then just use bracketed dollar signs as usual in LaTeX to switch to math environment, if you want to use exponential notation for example.

(3) I did understand the restriction to summands less than $p_{k+1}^2$ as being important to what you’re trying to do. What I am not understanding are arguments meant to support the italicized portion below

if $S_0(k)$ is not complete (re lemma 9.6 about unit pairsums covering $E(k)$ the set of even residues mod $p^k$) due to no prime-pair available, such missing $2n$ cannot be implemented by another composite pair or prime-and- composite pair, since they are too large, namely beyond the smallest composite $p_{k+1}^2$.

That is, I understand why you want to rule out such pairs, but I don’t understand how you rule them out.

(4) Let me ask again: isn’t the $p^k$ in

$G_1(k+1) \equiv G_1(k) \mod p^k$

a typo? Here I understand $G_1(k)$ to mean the group of units modulo $m_k$, the product of the first $k$ primes $p_1 = 2, \ldots, p_k$, and I interpret the congruence as intending to say that the reduction map

$\phi: \mathbb{Z}_{m_{k+1}} \to \mathbb{Z}_{m_k}$

maps units in $G_1(k+1)$ to $G_1(k)$, and also that if $\phi(u) = \phi(v)$, then $u \equiv v \mod p_{k+1}$ (not $p^k$; I don’t even know what “$p$” would be precisely).

I also asked you a question about why you couldn’t end the proof of theorem 8.1 in your paper right after you mention that not(GC) implies a contradiction with theorem 6.1.

Incidentally, we can continue this discussion over email; you can write me at topologicalmusings[at]gmail[dot]com. The moderators of this blog may appreciate our taking a long drawn-out discussion elsewhere.

Posted by: Todd Trimble on July 7, 2009 6:24 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Hi Todd, I was just going to reply with an easy way to exclude one or two composite summands in a missing 2n, namely consider in the first line of the last paragraph of thm9.2 (GC), only 2n \lt p_{k+1} 2.

Your point (4): indeed the modulus is m_k.

The statement ” 2n in S’(k)= P(k)+P(k) ” says that 2n is a sum of two primes, while this is in contradiction to the start of the sentence “Now assume GC to fail for some 2n in …” It should be:
Now assume GC to fail for some 2n \lt p_{k+1) exp 2. to exclude composite summands. Then S_0(k) is incomplete etc…

I also don’t like to continue here with further details. In fact I hope you understand the proof idea now, so we can stop this discussion. The best would be to wait for a half-price softcover version, which I pleaded for at Springer. It ‘only’ takes them to de-classify the book as a library-book, for which they only make that hard-cover version with high price. I strongly requested my editor (who indeed is an engineer;-) to do this de-classification, because my (and I notice his) intention is to keep the book available to a larger public of mathmaticians *and* engineers, because of its inter-disciplanairy nature (hence also NOT sell in parts online!-)
Best regards, Nico.

Posted by: Nico Benschop on July 7, 2009 7:01 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

If you want to discontinue the discussion, that’s fine. I have an inkling of your proof idea, but that doesn’t mean I have any confidence at all that you have established your claims of having proved GC. On the contrary, the errors and overall lack of clarity in the writing makes me believe that you have managed to hide the truth from yourself.

Mathematics can be a cruel business, and requires relentless honesty with one’s self on matters of intellectual truth, and against wishful thinking. The sine qua non would seem to be an innate rigorous adherence to the truth even if it means being disappointed, as far preferable to life under a cloud of potential self-deception or living with hidden doubts. I wish you good luck.

Posted by: Todd Trimble on July 7, 2009 8:28 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Todd wrote:

The moderators of this blog may appreciate our taking a long drawn-out discussion elsewhere.

Actually, the current style of conversation you’re having with Nico is welcome here: you are asking him questions about his argument, and he is answering them. It’s good to have this stuff on the public record. If you wind up believing Benschop’s proof, I’ll call the New York Times. If not… I won’t.

My impatience kicks in only when people start saying things like “Claiming to prove a major result out of the blue is what cranks do” or “I’ve seen enough denigrating stuff thrown at me the past 10 years… that I grew a hard skin”. This sort of wheel-spinning is incredibly common on the newsgroups, and pretty tiresome.

Posted by: John Baez on July 7, 2009 7:48 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Sorry John, I just posted a comment which may seem like more wheel-spinning. I guess it was meant as a coda to the fact that Nico would rather not discuss his alleged proof, apparently not even privately. Which is too bad.

I’ll be happy to let everyone know if he convinces me after all.

Posted by: Todd Trimble on July 7, 2009 8:36 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Todd, I can view the whole book here:

This is probably because my university has a subscription to Springerlink, so it’s probably worth checking whether you have some access to it via yours.

Posted by: Tom Ellis on July 7, 2009 8:14 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Not every good mathematician is affiliated to a university. This is one reason why people who are serious about math make it freely available, for example on the arXiv.

Posted by: John Baez on July 8, 2009 1:45 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

John Baez (8 jul 2009) wrote:
“Not every good mathematician is affiliated to a university. This is one reason why people who are serious about math make it freely available, for example on the arXiv.”

NB: For those interested in an elementary proof of Goldbach’s Conjecture (11 pgs) see http://de.arxiv.org/abs/math.GM/0103091 (v5)
And if you agree with the (residue-and-carry) approach, please let me know at nfbenschop@onsbrabantnet.nl — Springer Verlag will, with such support if it comes from an expert in this field, republish the book including Chapter 9 (containing the above Goldbach proof).

Posted by: Nico Benschop on November 3, 2009 8:49 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

So what’s the outcome? Benschop’s proof is invalidated or not?

Posted by: Anon on July 13, 2009 1:48 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Some mathematicians are still working on it for a second opinion (among others Todd Trimble in this column). A small correction, upon his remark, is made in thm(9.2)- in the sentence starting with “Now assume GC to fail for some 2n ….”. This version can (temporarily) be found on my homepage
http://home.claranet.nl/users/benschop/ngb0203.pdf – NB

Posted by: Nico Benschop on July 16, 2009 10:17 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

http://home.claranet.nl/users/benschop/ngb0907.pdf

Posted by: Nico Benschop on July 29, 2009 9:03 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Whether Mr. Benschop’s alleged proof of the Goldbach Conjecture has been invalidated may depend on who you ask. If you were to ask Robin Chapman, then I believe it is likely that he would reply that he has invalidated the proof put forward in the arXiv article, which Nico has said does not differ essentially from the proof given in his book. (See Chapman’s statement following the sci.math.research posting where the publication of Benschop’s book was announced.)

For some reason, some people seem to be looking to me to make a pronouncement. Unfortunately, I haven’t found enough time to examine the paper in detail. So I can’t point yet to a specific statement that I think is demonstrably false, but this doesn’t mean that I believe that Nico’s arguments are rigorous and/or free of gaps. Indeed, I personally find many of his arguments quite difficult to follow. In some such cases I’ve checked to see whether I could at least construct proofs for myself of certain assertions, and this has met with partial, but needless to say not complete success.

I won’t make any promises whether or when I’ll get around to forming an opinion on this paper which I think I can back up rigorously (although I’ll report back here if and when I do). As I’ve said before, life is short, and alleged elementary proofs of Fermat’s Last Theorem, Goldbach’s Conjecture, and so on are pretty far removed from my current interests. Seeing whether I can spot a flaw in this attempt would have to be considered a diversion for me, with no fixed time frame.

I assume Nico will not mind my saying that according to him, a “well-known Dutch number theorist” is also looking at his paper, for a second opinion.

Posted by: Todd Trimble on July 16, 2009 4:34 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Todd, a suggestion: I believe that an elementary* proof of Case I of FLT for p=7 would already be original. (Case I means that p doesn’t divide x, y or z.) It might be useful to not bother verifying any results which are obvious in this case. (For example, there is a lengthy discussion on usenet as to whether Nico has proved that p^3 never divides 2^p-2. This is obvious when p=7!)

Note that there are 7-adic solutions to Fermat’s equation: If A+B=C with A=30 mod 49, B=18 mod 49 and C=48 mod 49, then A, B and C have seventh roots in the 7-adics.

*By elementary, I basically mean one that uses neither cyclotomic fields, nor elliptic curves.

Posted by: David Speyer on July 16, 2009 11:07 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

David Speyer wrote: ” I believe that an elementary* proof of Case I of FLT for p=7 would already be original.”

NB: That’s where I indeed started in 1994, noticing that the cubic roots of 1 mod 49 were three 7-th powers adding to zero.
This holds for the cubic roots of 1 mod p exp k (k \gt 1, p == 1 mod 6). And for some (rare) p == -1 mod 6 there are (non-cubic) solutions that form a triplet as follws: (a+1)b == (b+1)c == (c+1)a == -1 with abc ==1 mod p exp k, starting at p=59. This elementary proof was published in Nov’2005 in the Acta Mathematica of the Univ. Bratislava. I made six announcements (one each 7-th of the month from oct’2006 till march 2007) in sci.math, with no noticable response - except one from DF who called for a boycot of U.Bratislava. – NB

http://www.emis.de/journals/AMUC/_vol74n2.html (p169-184)

Posted by: Nico Benschop on July 17, 2009 10:55 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

That's a very nice version for onscreen reading that they have there. (My only complaint is that they call the other version for ‘Download’ instead of what it really is, for ‘Printing’. I have instead downloaded the ‘Version to Read’, since I intend to read it offline but on screen and not print it.)

I'm almost inclined to read it now and try to understand the argument just for $p = 7$, since that should be simpler than the general argument but would already be new. (I say ‘almost inclined’ since I have a lot of other stuff on my plate, but maybe …).

Posted by: Toby Bartels on July 18, 2009 1:16 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Oh, just noticed that you are reading the Goldbach proof, not the FLT one. I’ll see if I can think of a similar simplification there.

Posted by: David Speyer on July 16, 2009 11:09 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

That’s okay, David – any suggestions you have, whether directed to the claimed proof of FLT or that of GC, are very welcome. You made a good point about FLT which hadn’t occurred to me, actually (was it Kummer who first established $p = 7$ for FLT, based on arithmetic of the cyclotomic field generated by seventh roots of unity?).

Posted by: Todd Trimble on July 17, 2009 4:48 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

“Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals” by Ivancevic and Ivancevic?

Seems to be a thousand pages of copied material. Though they often add references to copied passages. Actually I haven’t yet found a lot of sentences (in their book or in their articles) which do not give any hits on google (even Wikipedia shows up!).

Posted by: anm on July 14, 2009 1:25 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The first 171 pages of the book contain no proofs (I scanned the pdf file for the word “proof”). The book is not wrong! :)

Posted by: Eugene Lerman on July 14, 2009 4:21 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Plenty of “Proofs” around. None of then have proven to be very convincing though. “Proof” does not mean proven in this context and Springer have used the word correctly. Though the word is misleading it is the correct term for mathematical arguments of this nature.

Posted by: LaLa on August 2, 2009 8:54 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

NEWS FLASH!

The link to Benschop’s book on Springer’s website no longer works!

Posted by: John Baez on August 3, 2009 10:37 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Even more interestingly, searching for Benschop or the book title or the isbn on the Springer website gives no result. Any trace of the book has vanished.

A nice Stalinist touch! Must have been the Stasi sleeper cells at Springer that just woke up.

Posted by: Maarten Bergvelt on August 4, 2009 3:08 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Amazon still lists it.

Posted by: Greg Egan on August 4, 2009 5:01 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Buy it while you still can! It may become a collector’s item.

Or maybe Springer will continue to sell it while trying to minimize its visibility. They do have a contract with Benschop, after all…

… though authors usually forget to check whether the contract requires the publisher to keep the book in print.

The image of the book cover that graces this blog entry is still available on Springer’s website. Any bets on how long it takes for Springer to make it disappear, after someone there reads this?

Posted by: John Baez on August 4, 2009 7:20 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Dear John,

At least for its monographs (as this one) Springer employs print-on-demand, so it is no problem to make a correction in the book (Ch.9: Goldbach proof). In this case everyone involved agreed to disable the book’s availability untill such correction is done (with well-known math reviewer’s input). Sorry for the delay;-) Progress can be followed on http://home.claranet.nl/users/benschop/ng-abstr.htm

Posted by: Nico Benschop on August 4, 2009 9:16 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

Many years ago I received a manuscript for this book for consideration as a volume in my book series “Mathematics and its Applications” (Kluwer Acad. Publ. at that time).

I took a serious look at it and concluded it was complete nonsense and rejected it. I am still of that opinion.

Posted by: Michiel Hazewinkel on December 23, 2009 6:29 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

WoW, the voice of Authority who knows it all. Do you include, next to the present version of Goldbach’s Conjecture proof (chapter 9) the other 10 chapters of the book - which you apparently claim to have seen? Six of which are published in reputable journals and/or conferences: from state machine structure (the 5 basic components and the ways to couple them), Boolean Logic implementation (including fault-tolerant logic design) and Arithmetic (incl. a European Esprit project for a 32 bit logarithmetic unit on silicon). Maybe it’s useful for you to check the arXiv at
http://de.arxiv.org/abs/math/0103091 (v5)
to see the Goldbach Conjecture proof. Don’t be embarrassed to report if it improved (possibly even due to your suggestions;-)

Best greetings and wishes for the New Year.

Posted by: Nico Benschop on December 27, 2009 9:06 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

For the record.
Several years ago, possibly as many as 10, I received texts from Nico Benschop on FLT and GC with a view towards publication in my book series ‘Mathematics and its applications’ (or my journal ‘Acta appl. math.’). I found many statements to obscure to understand. One glaring mistake concerned the structure of the groups of units on the rings of integers modulo a prime power. I wrote him a long letter rejecting the material and pointing out some of the mistakes.

Posted by: Michiel Hazewinkel on December 26, 2009 1:38 PM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

By the way: the elementary proof of FLT that you mention as being ‘complete nonsense’ is published in the Nov.2005 issue of Acta Mathematica of Univ-Bratislava: http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html

Posted by: Nico Benschop on December 28, 2009 10:01 AM | Permalink | Reply to this

### Re: Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture

The Springer book referred to, published April 2009, was put out of print a few months later (Aug. 2009) without giving a reason.
I re-obtained my copyright to the book back from Springer in 2010. It is now available (with an improved chapter 9: Goldbach elementary proof), under self-publication at abc.nl (American Book Center, Amsterdam) for 19.50 euro + postage (vs. the original 96 euro at Springer), see my homepage http://home.claranet.nl/users/benschop — including two reviews, one from ACM Computer Reviews, and one from Zentralblatt.Math.

Posted by: Nico Benschop on March 7, 2011 11:33 AM | Permalink | Reply to this

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