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March 14, 2020

The Hardest Math Problem

Posted by John Baez

Not about coronavirus… just to cheer you up:

Puzzle. What math problem has taken the longest to be solved? It could be one that’s solved now, or one that’s still unsolved.

Let’s start by looking at one candidate question. Can you square the circle with compass and straightedge? After this question became popular among mathematicians, it took at least 2296 years to answer it!

It’s often hard to find when a classic math problem was first posed. As for squaring the circle, MacTutor traces it back to before Aristophanes’ wacky comedy The Birds:

The first mathematician who is on record as having attempted to square the circle is Anaxagoras. Plutarch, in his work On Exile which was written in the first century AD, says:

“There is no place that can take away the happiness of a man, nor yet his virtue or wisdom. Anaxagoras, indeed, wrote on the squaring of the circle while in prison.”

Now the problem must have become quite popular shortly after this, not just among a small number of mathematicians, but quite widely, since there is a reference to it in a play The Birds written by Aristophanes in about 414 BC. Two characters are speaking, Meton is the astronomer.

Meton: I propose to survey the air for you: it will have to be marked out in acres.

Peisthetaerus: Good lord, who do you think you are?

Meton: Who am I? Why Meton. THE Meton. Famous throughout the Hellenic world - you must have heard of my hydraulic clock at Colonus?

Peisthetaerus (eyeing Meton’s instruments): And what are these for?

Meton: Ah! These are my special rods for measuring the air. You see, the air is shaped - how shall I put it? - like a sort of extinguisher: so all I have to do is to attach this flexible rod at the upper extremity, take the compasses, insert the point here, and - you see what I mean?

Peisthetaerus: No.

Meton: Well I now apply the straight rod - so - thus squaring the circle: and there you are. In the centre you have your market place: straight streets leading into it, from here, from here, from here. Very much the same principle, really, as the rays of a star: the star itself is circular, but sends out straight rays in every direction.

Peisthetaerus: Brilliant - the man’s a Thales.

Now from this time the expression ‘circle-squarers’ came into usage and it was applied to someone who attempts the impossible. Indeed the Greeks invented a special word which meant ‘to busy oneself with the quadrature’. For references to squaring the circle to enter a popular play and to enter the Greek vocabulary in this way, there must have been much activity between the work of Anaxagoras and the writing of the play. Indeed we know of the work of a number of mathematicians on this problem during this period: Oenopides, Antiphon, Bryson, Hippocrates, and Hippias.

So, quite conservatively we can say that the squaring the circle was an open problem known to mathematicians since 414 BC. It was proved impossible by Lindemann in 1882, when he showed that e xe^x is transcendental for every nonzero algebraic number xx. Taking x=iπx = i \pi this implies that π\pi is transcendental, and thus cannot be constructed using straightedge and compass.

So, this problem took at least 1882 + 414 = 2296 years to settle!

Can you find one that took longer to solve? It’s often hard to find when ancient problems were first posed. There’s trisecting the angle and doubling the cube, the other two classic Greek geometry challenges. Trisecting the angle was proved impossible in 1836 or 1837 by Wantzel. So, it would have to have been posed at least before 460 BC to beat squaring the circle. I don’t know when people started wondering about it.

How about the question of whether there are infinitely many perfect numbers? This still hasn’t been solved, so it would only need to have to been posed before 276 BC to beat squaring the circle. This seems plausible, since Euclid proved that 2 p1(2 p1)2^{p−1}(2^p − 1) is an even perfect number whenever 2 p12^p - 1 is prime: it’s Prop. IX.36 in the Elements, which dates to 300 BC.

Alas, I don’t think Euclid’s Elements asks if there are infinitely many perfect numbers. But if Euclid wondered about this before writing the Elements, the question may have been open for at least 2020 + 300 = 2320 years!

Can you help me out here?

Posted at March 14, 2020 9:55 PM UTC

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Re: The Hardest Math Problem

One cannot look to how long a problem has been around before it has been solved to judge the difficulty of the problem, as there are entire fields of mathematics that were discovered/invented millennia after some of the above problems mentioned. If the ancient Greeks studied real analysis and category theory along with geometry and number theory we would have a better judge by time as to which problems are harder than which other problems.

Posted by: Madeleine Birchfield on March 15, 2020 5:39 AM | Permalink | Reply to this

Re: The Hardest Math Problem

One can imagine many different ways to measure how hard a math problem is. I’m just interested in which problem has taken the longest to solve because it’s something we can study rather easily, just by reading the history of math. I don’t claim this is the “correct” measure of hardness.

It could be that some problems have remained unsolved for a long time merely because they were obscure, not because they’re hard. But the famous Greek trio — doubling the cube, trisecting the angle and squaring the circle — called for the development of group theory, Galois theory, and (in the case of squaring the circle) complex analysis and the theory of symmetric functions.

The main difficulty with finding the math problem that was unsolved for the longest time seems to be that the origin of certain famous puzzles is obscure. When did someone first wonder if there were infinitely many twin primes, or perfect numbers — or if there exists an odd perfect number?

Posted by: John Baez on March 15, 2020 5:43 PM | Permalink | Reply to this

Re: The Hardest Math Problem

Yes, but the history of mathematics could just be a question of whether a mathematician is creative/inventive enough to conceive of new concepts and whether the mathematical community is willing to accept such new concepts, such as the decimal positional numeral system in ancient Greece instead of in medieval India, or the use of Clifford algebra in Newtonian mechanics and electromagnetism in the late 1870s instead of the 1960s. So if the ancient Greeks successfully formulated field theory and algebraic field extensions, then doubling the cube could be solved in a fairly short time in the alternate timeline, but it wouldn’t make doubling the cube an easier problem in the alternate timeline.

And my point was also that there might be problems in other fields of mathematics that mught be harder than those in integer number theory or geometry, such as the Riemann hypothesis in complex analysis, but cannot even be formulated by the Ancient Greeks because complex numbers were not known/invented until about 400 years ago.

Posted by: Madeleine Birchfield on March 15, 2020 7:51 PM | Permalink | Reply to this

Re: The Hardest Math Problem

Perhaps my remarks are about the title which implies a discussion about the difficulty of maths problems in the blog post rather than the number of years a maths problem has remained unsolved.

Posted by: Madeleine Birchfield on March 15, 2020 10:06 PM | Permalink | Reply to this

Re: The Hardest Math Problem

Side comment on doubling the cube and angle trisections: I haven’t looked into how whoever it was (Wentzel or someone) proved their impossibility using ruler and compass, but Galois groups and such is overkill. All one needs is to count dimensions of field extensions, and realize that a field of degree 33 over \mathbb{Q}, like [23]\mathbb{Q}[\sqrt[3]{2}] or [sin(π/9)]\mathbb{Q}[\sin(\pi/9)], can never be a subfield of any field of degree 2 n2^n (where a ruler-and-compass construction lives), because 33 doesn’t divide 2 n2^n.

Posted by: Todd Trimble on March 15, 2020 6:43 PM | Permalink | Reply to this

Re: The Hardest Math Problem

“Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas” by Pierre Wantzel.

Posted by: Madeleine Birchfield on March 15, 2020 7:58 PM | Permalink | Reply to this

Re: The Hardest Math Problem

The problems John Baez mentions are hard because a solution requires new fields of mathematics. This is what makes them hard.

Posted by: Jay Kangel on March 15, 2020 1:46 PM | Permalink | Reply to this

Re: The Hardest Math Problem

It’s likely that the Pythagoreans thought about perfect numbers. But since there’s not much to go on from them in written form, it’s anyone’s guess whether they asked if there were infinitely many.

The only thing about this from antiquity I was able to find is what’s described here from Maths History, about the writings of Nicomachus from about 100 AD (or 100 CE if you prefer). Over at History of Science and Mathematics Stack Exchange, user Conifold seems to write carefully about such matters.

I gather that proposing problems and conjectures as such didn’t play a huge role in the writing of mathematics in antiquity. [For example, Nicomachus was apparently more in the habit of proclaiming certain things as facts (some of which weren’t), rather than asking them as questions.] There is the famous exception of Archimedes and his Cattle Problem, which is put more in the way of a challenge, and which had a pretty long run before Amthor produced his solution in 1880 (I don’t know if they have a date for when Archimedes proposed the problem, but we could guess maybe 250 BCE, so that’s about 2130 years, give or take).

As for twin primes, the sources I see point to Polignac – from 1849. Not to some ancient Greek/Hellenic person posing the question, either explicitly or implicitly.

Posted by: Todd Trimble on March 16, 2020 3:18 AM | Permalink | Reply to this

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