Bounded Gaps Between Primes

May 14, 2013

Posted by Tom Leinster

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Whether we grow up to become category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers. It comes as no surprise then that a seminar given yesterday afternoon at Harvard by Yitang Zhang of the University of New Hampshire reporting on his new paper “Bounded gaps between primes” attracted a diverse audience. I don’t believe the paper is publicly available yet, but word on the street is that the referees at the Annals say it all checks out.

What follows is a summary of his presentation. Any errors should be ascribed to the ignorance of the transcriber (a category theorist, not an analytic number theorist) rather than to the author or his talk, which was lovely.

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Prime gaps

Let us write $p_1, p_2, \ldots$ for the primes in increasing cardinal order. We know of course that this list is countably infinite. A prime gap is an integer $p_{n+1}-p_n$ . The Prime Number Theorem tells us that $p_{n+1}-p_n$ is approximately $\log(p_n)$ as $n$ approaches infinity.

The twin primes conjecture, on the other hand asserts that

$\liminf_{n \to \infty} (p_{n+1}-p_n) =2$

i.e., that there are infinitely many pairs of twin primes for which the prime gap is just two. A generalization, attributed to Alphonse de Polignac, states that for any positive even integer, there are infinitely many prime gaps of that size. This conjecture has been neither proven nor disproven in any case. These conjectures are related to the Hardy-Littlewood conjecture about the distribution of prime constellations.

The strategy

The basic question is whether there exists some constant $C$ so that $p_{n+1}-p_n \lt C$ infinitely often. Now, for the first time, we know that the answer is yes…when $C = 7 \times 10^7$ .

Here is the basic proof strategy, supposedly familiar in analytic number theory. A subset $H = \{ h_1,\ldots, h_k \}$ of distinct natural numbers is admissible if for all primes $p$ the number of distinct residue classes modulo $p$ occupied by these numbers is less than $p$ . (For instance, taking $p=2$ , we see that the gaps between the $h_j$ must all be even.) If this condition were not satisfied, then it would not be possible for each element in a collection $\{ n + h_1,\ldots, n +h_k\}$ to be prime. Conversely, the Hardy-Littlewood conjecture contains the statement that for every admissible $H$ , there are infinitely many $n$ so that every element of the set $\{ n + h_1,\ldots, n +h_k\}$ is prime.

Let $\theta(n)$ denote the function that is $\log(n)$ when $n$ is prime and 0 otherwise. Fixing a large integer $x$ , let us write $n \sim x$ to mean $x$ ≤ $n \lt 2x$ . Suppose we have a positive real valued function $f$ —to be specified later—and consider two sums:

$S_1 = \sum_{n \sim x} f(n)$

$S_2 = \sum_{n \sim x} \biggl( \sum_{j=1}^k \theta(n+h_j)\biggr) f(n)$

Then if $S_2 \gt (\log 3x) S_1$ for some function $f$ it follows that $\sum_{j=1}^k \theta(n+h_j) \gt \log 3x$ for some $n \sim x$ (for any $x$ sufficiently large) which means that at least two terms in this sum are non-zero, i.e., that there are two indices $i$ and $j$ so that $n+h_i$ and $n+h_j$ are both prime. In this way we can identify bounded prime gaps.

Some details

The trick is to find an appropriate function $f$ . Previous work of Daniel Goldston, János Pintz, and Cem Yildirim suggests define $f(n) = \lambda(n)^2$ where

$\lambda(n) = \sum_{d \mid P(n), d \lt D} \mu(d) \Bigl(\log \Bigl(\frac{D}{d}\Bigr)\Bigr)^{k+\ell} \quad\quad P(n) = \prod_{j=1}^k(n+h_j)$

where $\ell \gt 0$ and $D$ is a power of $x$ .

Now think of the sum $S_2 - (\log 3x) S_1$ as a main term plus an error term. Taking $D = x^\vartheta$ with $\vartheta \lt \frac{1}{4}$ , the main term is negative, which won’t do. When $\vartheta = \frac{1}{4} + \omega$ the main term is okay but the question remains how to bound the error term.

Zhang’s work

Zhang’s idea is related to work of Enrico Bombieri, John Friedlander, and Henryk Iwaniec. Let $\vartheta = \frac{1}{4} + \omega$ where $\omega = \frac{1}{1168}$ (which is “small but bigger than $\epsilon$ ”). Then define $\lambda(n)$ using the same formula as before but with an additional condition on the index $d$ , namely that $d$ divides the product of the primes less that $x^{\omega}$ . In other words, we only sum over square-free $d$ with small prime factors.

The point is that when $d$ is not too small (say $d \gt x^{1/3}$ ) then $d$ has lots of factors. If $d = p_1\cdots p_b$ and $R \lt d$ there is some $a$ so that $r= p_1\cdots p_a \lt R$ and $p_1\cdots p_{a+1} \gt R$ . This gives a factorization $d = r q$ with $R/ x^\omega \lt r \lt R$ which we can use to break the sum over $d$ into two sums (over $r$ and over $q$ ) which are then handled using techniques whose names I didn’t recognize.

On the size of the bound

You might be wondering where the number 70 million comes from. This is related to the $k$ in the admissible set. (My notes say $k = 3.5 \times 10^6$ but maybe it should be $k = 3.5 \times 10^7$ .) The point is that $k$ needs to be large enough so that the change brought about by the extra condition that $d$ is square free with small prime factors is negligible. But Zhang believes that his techniques have not yet been optimized and that smaller bounds will soon be possible.

Posted at May 14, 2013 8:44 PM UTC

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Re: Bounded Gaps Between Primes

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Nice summary!

The Prime Number Theorem tells us that $p_{n+1}−p_n$ is approximately $\log(p_n)$ as $n$ approaches infinity.

Of course, it’s better to say $p_{n+1}−p_n$ is $\log(p_n)$ on average as $n$ approaches infinity.

By the way, in 2004, Daniel Goldston, János Pintz and Cem Yıldırım were able to show that there are infinitely many pairs of primes at most 16 apart… if something called the Elliott–Halberstam conjecture is true.

This is a really nice expository article about the whole issue:

• K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım.

Posted by: John Baez on May 15, 2013 3:38 PM | Permalink | Reply to this

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Yes, of course! That was sloppy of me. Thanks.

Posted by: Emily Riehl on May 15, 2013 4:09 PM | Permalink | Reply to this

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Thanks so much for this nice summary! It makes me feel like I can almost understand it.

In other news I see this arxiv paper, claiming that the ternary Goldbach conjecture has been proven: (by H.A. Helfgott, Ecole Normale Superior) Major arcs for Goldbach’s theorem .

If anyone can summarize those 131 pages that would be awesome!

Posted by: stefan on May 15, 2013 4:24 PM | Permalink | Reply to this

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Stefan wrote:

If anyone can summarize those 131 pages that would be awesome!

I can’t do that, but here’s some chat. On Google+, Terence Tao wrote:

Busy day in analytic number theory; Harald Helfgott has complemented his previous paper (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes. (This improves upon a result of mine from last year showing that such numbers are the sum of five or fewer primes, though at the cost of a significantly lengthier argument.) As with virtually all successful partial results on the Goldbach problem, the argument proceeds by the Hardy-Littlewood-Vinogradov circle method; the challenge is to make all the estimates completely effective and to optimise all parameters (which, among other things, requires a certain amount of computer-assisted computation).

I wrote:

Today +Harald Helfgott publicized his proof of the odd Goldbach conjecture:

Every odd number greater than 5 can be expressed as the sum of 3 primes.

I’m optimistic that it’s correct, not because I understand it, but because Helfgott has a good track record and +Terence Tao, an expert on these matters, sounds optimistic.

Actually Helfgott’s proof only works for odd numbers greater than 10^30. But the result has already been checked by computer for odd numbers smaller than this!

Before Helfgott proved it for odd numbers greater than $10^{30}$ , the odd Goldbach conjecture was known to be true for numbers greater than $e^{3100}$ . This meant it could in principle be checked by a computer… but not in practice now, because that number was too big.

Interestingly, Helfgott’s proof for odd numbers greater than $10^{30}$ also relies on computer calculations! As part of the work, David J. Platt needed to check hundreds of thousands of facts that would be true if the Generalized Riemann Hypothesis holds. I don’t think I want to explain this, since you can see the basic idea here:

• Generalized Riemann Hypothesis, Wikipedia.

But, briefly, this hypothesis says that all the zeros of certain functions called Dirichlet L-functions that lie in a certain strip of the complex plane actually lie on a certain line.

In a conversation last May here on G+, Helfgott said:

Very informally and off the strictest record […] friends say […] “this is now done modulo 2-years-and-£100000 worth of computer time”. I think this is very roughly right. It is of course possible that the result will be improved or complemented, either by myself or by others, before anybody puts in very serious computer resources to the task.

It seems that Platt did the calculations sooner and more cleverly than expected!

In case you’re wondering, the odd Goldbach conjecture has long been considered much easier than the original Goldbach conjecture: every even number greater than 2 can be expressed as the sum of 2 primes. Tao says brand-new insights would be needed to crack this one.

Harald Helfgott replied:

Hi John -

Actually, what happened is that I improved both my minor arcs results (hence the new version of that paper posted yesterday) and my ideas towards major arc results (included in the new paper, Major Arcs…, also posted yesterday). This meant that the computations that had to be done went further by only a factor of 2 or so than those that Platt had already done for his thesis. (This translates into a factor of 8 or so in computer time.)

I should also say that Platt is, of course (as you say), clever, and a pleasure to work with. Also, our joint efforts to get people to give us more computer time bore fruit thanks to the generosity of several institutions and individuals.

Let me add that my proof really works starting at $10^27$ or so, or, with some modifications, perhaps even a little below that. I’ve assumed $n \ge 10^30$ , which is both more than I need and less than what has been checked, so as to give me a wide berth in case of minor slips.

Posted by: John Baez on May 15, 2013 8:28 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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Thank you, John! This is great stuff, and I’ll check back at the Google+ conversation for any future comments.

Posted by: stefan on May 22, 2013 6:58 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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If we follow Elkies

The Harvard math curriculum leans heavily towards the systematic, theory-building style; analytic number theory as usually practiced falls in the problem-solving camp. This is probably why, despite its illustrious history (Euclid, Euler, Riemann, Selberg, … ) and present-day vitality, analytic number theory has rarely been taught here. … Now we shall see that there is more to analytic number theory than a bag of unrelated ad-hoc tricks, but it is true that partisans of contravariant functors, adèlic tangent sheaves, and étale cohomology will not find them in the present course. Still, even ardent structuralists can benefit from this course…. An ambitious theory-builder should regard the absence thus far of a Grand Unified Theory of analytic number theory not as an insult but as a challenge. Both machinery- and problem-motivated mathematicians should note that some of the more exciting recent work in number theory depends critically on symbiosis between the two styles of mathematics.

Is there any whiff of such symbiosis in this latest work?

Posted by: David Corfield on May 16, 2013 8:27 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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I have a question. maybe it’s fool, but every people are sometimes fool in the front of God.So please don’t laugh at me. if p(n+1) - p(n) < C => 1-p(n)/p(n+1) < C/p(n+1). we know that lim(C/p(n+1)) = 0, but lim(1-p(n)/p(n+1)) = 1 - lim(p(n)/p(n+1)) >0, so left > 0 and right =0, this inequation (left < right) is not correct =>p(n+1) - p(n) < C is not correct.

Posted by: freepublic on May 16, 2013 11:08 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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Let me pretend for a moment that $p_n$ is shorthand for the function $n \times (C-\epsilon)$ . Then $p_{n+1} - p_n \lt C$ and as you observe $1 - \frac{p_n}{p_{n+1}} \lt \frac{C}{p_{n+1}}$ . The limit as $n \to \infty$ of the right hand side is zero but so is the left! Note that $\frac{p_n}{p_{n+1}} = \frac{n}{n+1}$ .

It is a little more complicated to phrase limiting statements in our context, when the $p_n$ are primes. To say that there are infinitely many prime gaps less than $C$ is to say that the limsup of $p_{n+1}-p_n$ is less that $C$ as $n \to \infty$ .

The prime number theorem, which I slightly misquoted above, says that the function $\pi(n)$ which counts the number of primes less than $n$ is asymptotically equal to $\frac{n}{\log(n)}$ , meaning if you take the limit as $n \to \infty$ of the ratio of these functions you get one. As John points out above, the correct way to express this colloquially is to say that the prime gaps $p_{n+1}-p_n$ are $\log(p_n)$ on average as $n$ gets large.

Posted by: Emily Riehl on May 16, 2013 3:55 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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thanks for your explanation. I have some confusion still. if p(n)=n*(C-e), lim(p(n)/p(n+1))=lim(n/(n+1))=1, but left = 1-1=0,right=0,left10^C, log(p(n))>C. yes, if it’s on average, there are some gaps less than C, but more gaps above C. the ratio become smaller and smaller.when p(n) =infinity, the ratio become zero. no gap is less than C.

Posted by: freepublic on May 16, 2013 5:27 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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Sorry, I accidentally deleted some comments here. Freepublic posted two identical copies of his/her “thanks for your explanation…” comment, one of which I deleted — but an unintended side-effect was that replies to that comment also got removed.

Posted by: Tom Leinster on May 17, 2013 1:50 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

never mind, because i have some print error, then post twice.

my opinion is if construct a regular sequence X = x(n+1)-x(n) = log(n), the total of x whoese gaps
another way, we can divide axis like this:0____10^C______infinity, we can move all of p(n) whoes gaps are less than C to [0,10^C], and all of p(n) whoes gaps are bigger than C to [10^C,infinity).because the p(n+1)-p(n)=log(n) on average. then the total primes of [0,10^C] eual to total primes of [10^C,infinity). but how could [0,10^C] contain infinite primes?

Posted by: freepublic on May 18, 2013 3:23 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

if construct a regular sequence X = x(n+1)-x(n) = log(n), the total of x whose gaps are less than C is finite. X can transform to prime sequence by order and value change.

Posted by: freepublic on May 18, 2013 3:38 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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you have mentioned that “A subset H={h1,…,hk} of distinct natural numbers is admissible if for all primes p the number of distinct residue classes modulo p occupied by these numbers is less than p. (For instance, taking p=2, we see that the gaps between the hj must all be even.) If this condition were not satisfied, then it would not be possible for each element in a collection {n+h1,…,n+hk} to be prime. ” could you give me a example to understand. does it mean that p=2, H={1,3,5,7,9,..} hk=?

Posted by: freepublic on May 25, 2013 6:33 PM | Permalink | Reply to this

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Hi! No, {1,3,5,etc.} would not be admissible, as for p=3 all modulo classes are occupied: 1 is 1 mod 3, 3 is 0 mod 3, 5 is 2 mod 3. What this means is that whatever number you add {1,3,5,etc.} to, one of the resulting numbers is going to be 0 mod 3, i.e. a multiple of 3. If you try 10, you get 11, 13, 15 (boom!). So if the set H is not admissible, you know that (at least for large enough n that you add) you will get a non-prime number. I find it quite remarkable that this modest necessary condition should also be a sufficient one for getting infinitely many prime constellations, should the mentioned conjecture be true.

Posted by: Edwin Steiner on May 27, 2013 7:36 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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does it mean that, H={1,2}, H=(1,2,3,4} are admussible. becuase 3>2, 5>3. if p=2, H={1}, there are infinite n, {2+1}, {4+1}, {6+1},{10+1}… are primes?

Posted by: freepublic on May 28, 2013 11:58 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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5>4, fix a write error

Posted by: freepublic on May 28, 2013 11:59 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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It does not work like “pick a prime p, then find an admissible set H for p”. To be admissible, H must satisfy the condition for all primes p. (Obviously, if you have k elements in H you need only check p <= k, because you cannot occupy >k modulo classes with k numbers.) So {1,2} is not admissible as it is {0,1} mod 2. {1} is admissible and yields the trivial constellations of a single prime each. See http://mathworld.wolfram.com/PrimeConstellation.html for valid examples.

Posted by: Edwin Steiner on May 28, 2013 6:58 PM | Permalink | Reply to this

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This is another explanation including references: http://primes.utm.edu/glossary/xpage/PrimeConstellation.html

Posted by: Edwin Steiner on May 28, 2013 7:09 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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thanks for your example. I can understand Zhang’s strategy now. and investigate more details.

Posted by: freepublic on May 29, 2013 4:29 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

“Whether we grow up to become category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers.”

I highly disagree with this. I have interest in about 90% of all mathematics and was never interested in elementary features (chanting?) of prime numbers (though I appreciate nowdays large parts of advanced number theory, because I learned of its relation to other things in mathematics, like algebraic geometry), partly because the community of followers likes to emphasis on tricks. In fact, as a school boy, although I was very good in mathematics, I did not even consider ever becoming a mathematician until I learned about areas of mathematics which do not have references to calculation with numbers (and with indeterminates with the same rules as for numbers). I find quite annoying when people argue (without any arguments) that the usual numbers are the basic and natural, god given subject, unlike other areas of mathematics which are supposedly artificial and so on. This kind of hi brow aristocracy within mathematics is not for any mathematician to be proud of.

Posted by: Zoran Skoda on May 16, 2013 2:54 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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I agree with Zoran that the natural god-given virtue of prime numbers is greatly exaggerated. It is true we all spent a few pleasant hours in 5th or 6th grade factoring numbers like 1250 and 210 but never numbers like 91 or 391. After that it gets boring pretty quickly.

Probably the interest of any area of mathematics lies in how hard the unsolved problems are. Twin primes for example are only interesting because it is so very hard to prove things about them. The proofs such as Zhang’s proof are amazingly intricate machines made out of parts with their own history and connections to other areas of mathematics. All of this effort is more interesting than the object it is directed toward.

Posted by: Daniel Goldston on May 18, 2013 3:15 AM | Permalink | Reply to this

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It’s very interesting to read such a statement by a researcher. What also puzzles me in popular texts is when they go on about the amazing apparent randomness of the primes. In my (non-expert) view, the sieve of Eratosthenes is basically a systematic way of removing all regularity (I realize the tension in this statement), so I do not find it surprising that the primes show pseudo-random properties. What I find fascinating about primes is the mysterious connections between addition and multiplication. In my intuition, primes are natural inhabitants of the “multiplicative world”, where they are simple, and it is striking that they turn out to be so exceedingly complicated when looked at in the “additive world” (regarding sums of primes, gaps, etc.).

Posted by: Edwin Steiner on June 1, 2013 11:37 AM | Permalink | Reply to this

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Surely the sieve of Eratosthenes only remove “zero-based” regularity?

Posted by: Tom Ellis on June 1, 2013 6:34 PM | Permalink | Reply to this

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I see what you are getting at, but in fact my statement was not meant as something that could be made precise. It is just my reaction on a naive level to the usual popular exposition of the primes, where they first explain how primes are defined and then they show the pattern of the first 100 or so and say “Look how randomly they are scattered! Isn’t that amazing?”, and I think to myself “Well, in the step before you just removed any obvious regularity, so what’s the big deal?”. There may well be some truly amazing pseudo-random properties that are just not mentioned in these texts. The Chebyshev bias is an example that there are at least interesting statistical deviations from pseudo-randomness.

Posted by: Edwin Steiner on June 2, 2013 10:28 AM | Permalink | Reply to this

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She was merely pointing out the fact that prime numbers are fundamental in mathematics.

Posted by: timur on May 23, 2013 4:26 PM | Permalink | Reply to this

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The preprint is available from the Annals. You or your institution need to be a subscriber.

Oh, and Emily, $k=3.5\times 10^6$ is correct.

Posted by: David Roberts on May 21, 2013 11:39 PM | Permalink | Reply to this

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The k=3500000 is correct: http://www.wolframalpha.com/input/?i=PrimePi%28n%29-+PrimePi%283500000%29++%3D+%283500000%29

Posted by: Ali on May 28, 2013 7:10 PM | Permalink | Reply to this

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I can’t say I saw the real point of primes until I came across algebraic number theory and algebraic geometry. Its importance then became much more vivid.

Certainly its true that its a profound enough, but also simple enough idea that even most educated laymen will know what it is about vaguely. Which is why the media seem inordinately keen on them.

Posted by: mozibur ullah on May 28, 2013 10:30 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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In case anyone isn’t following, there’s discussion at Secret Blogging Seminar where by reasonably simple reasoning the bound is down to 57 554 086.

However the key number to reduce is $k_0 = 3.5\times 10^6$ , which hasn’t been touched; all reductions have been in how the bound $k_0$ has been applied.

Posted by: David Roberts on May 31, 2013 9:37 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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I do not quite see that there is two primes n+hi and n+hj. I could only see that there are two primes n1 +hi and n2 +hj with !n1 -n2! < x.

Also, how do we know that \pi (70000000)-\pi (3500000)>3500000?

Posted by: Daniel on May 31, 2013 6:46 PM | Permalink | Reply to this

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I think that I got the first question. And am still trying to find the answer of the second. Thanks.

Posted by: Daniel on May 31, 2013 11:38 PM | Permalink | Reply to this

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Also, how do we know that $\pi (70000000)-\pi (3500000) \gt 3500000$ ?

There are probably a bunch of ways. Looking to Google and Wikipedia for help, one can look up bounds on the prime-counting function, where one finds inequalities such as

$\frac{x}{\ln x}\left(1+\frac{1}{\ln x}\right) \lt \pi(x) \lt \frac{x}{\ln x}\left(1+\frac{1}{\ln x}+\frac{2.51}{(\ln x)^2}\right).$

Call the analytic function on the left $F_1(x)$ and the analytic function on the right $F_2(x)$ . According to these inequalities, we have

$\pi(70000000) - \pi(3500000) \gt F_1(70000000) - F_2(3500000) \gt 4089630 - 250259 = 3839371$

and probably cruder inequalities would work as well, but this gets the job done.

Posted by: Todd Trimble on June 1, 2013 2:54 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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in zone [x,3x], when x trends to infinity.zone becomes [infinity,3*infinity].then p(n+1)-pn=infinity, it satisfy the strategy condition. we think there are 2 primes in zone [x,3x]. but p(n+1)-pn=infinity, which is equivalent to that p(n+1) is not existing.actually, there is only 1 prime in [x,3x], but we think mistakenly there are 2 primes. so pn-pn=0

Posted by: freepublic on July 12, 2013 8:15 AM | Permalink | Reply to this

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I see they’ve got the bound down from 70 million to 285,232.

Posted by: David Corfield on June 10, 2013 12:09 PM | Permalink | Reply to this

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And now the gap is down to 60 744, using a $k_0=6329$ (down from 3.5 million in Emily’s post),

Posted by: David Roberts on June 17, 2013 3:44 AM | Permalink | Reply to this

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This morning, I went to an excellent talk on this by Ben Green. One aspect that particularly caught my attention was admissible sets.

Emily already said this in her post above, but I’ll repeat it in Ben’s vivid way. He said: a set $\{h_1, \ldots, h_k\}$ is admissible if there is a chance that there might be infinitely many numbers $n$ such that $n + h_1, \ldots, n + h_k$ are all prime.

The conjecture is then: for any admissible set, there really are infinitely many numbers $n$ such that $n + h_1, \ldots, n + h_k$ are all prime.

(Of course, he made the phrase “there is a chance” precise: as Emily said, it means that for any prime $p$ , the image of $\{h_1, \ldots, h_k\}$ in $\mathbb{Z}/p\mathbb{Z}$ is a proper subset. Actually, it doesn’t matter whether we say “prime $p$ ” or “number $p \geq 2$ ” here.)

The twin prime conjecture is the case $\{h_1, h_2\} = \{0, 2\}$ of the conjecture above. One thing neither Emily nor Ben pointed out is that the conjecture above would also imply the Green–Tao theorem, that there are arbitrarily long arithmetic progressions of primes.

To see this, just note that for any $n \geq 1$ , the sequence $1\cdot n!,\quad 2\cdot n!,\quad \ldots,\quad n\cdot n!$ is admissible. Indeed, for a prime $p$ greater than $n$ , obviously these can’t represent all the residue classes mod $p$ ; and for a prime $p$ less than or equal to $n$ , these are all zero mod $p$ , so again don’t represent all the residue classes. So the conjecture above implies that there are infinitely many arithmetic progressions of length $n$ and step size $n!$ .

But I’m most interested in something more general. The conjecture above (does it have a name?) is of the form “if there’s no obvious reason for there not to be infinitely many primes satisfying such-and-such, then there really are infinitely many primes satisfying such-and-such”. Here “obvious reason” refers to very simple considerations mod $p$ .

Are there more general statements of this type? I mean, some precise conjecture of the form “if it’s not obviously false that there are infinitely primes satisfying such-and-such, it’s true”?

Posted by: Tom Leinster on June 20, 2013 1:43 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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The most general conjecture which I know along these lines is Schinzel’s hypothesis H: For any polynomials $f_1(n)$ , $f_2(n)$ , … $f_r(n)$ , if there is no “obvious” obstacle to all the $f_i$ taking prime values simultaneously, then they do so infinitely often.

You want to be a little careful with guesses like this. I would guess that there are only finitely many primes of the form $2^{n^2}+3$ , even though there is no obvious obstruction, because the “probability that $t$ is prime” is $1/\log t$ and $\sum 1/\log(2^{n^2}+3)$ converges.

A little more subtly, there is no obvious obstruction to $2^n+1$ being prime, and $\sum 1/\log(2^n+1)$ converges. However, $2^n+1$ being prime implies that $n$ is a power of $2$ , and $\sum 1/\log(2^{2^k}+1)$ converges. So I would guess only finitely many primes of the form $2^n+1$ , but for a reason which seems hard to make into a general condition.

Posted by: David Speyer on July 12, 2013 6:29 PM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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Thanks so much for this nice summary! It makes me feel like I can almost understand it. I’ll check back at the Google+ conversation for any future comments. I can understand Zhang’s strategy now. and investigate more details. Good job!

Posted by: Melissa on April 14, 2014 8:49 AM | Permalink | Reply to this

Re: Bounded Gaps Between Primes

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Yitang Zhang awarded 2014 McArthur Fellowship

Posted by: RodMcGuire on September 20, 2014 11:36 PM | Permalink | Reply to this

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May 14, 2013