The Hardest Math Problem

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 2295 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^ix is transcendental for every nonzero algebraic number x. Taking x = iπ this implies that π is transcendental, and thus cannot be constructed using straightedge and compass.

So, this problem took at least 1882 + 414 – 1 = 2295 years to settle!

Puzzle. Why did I subtract 1 here?

But here’s what I really want to know: can you find a math problem that took longer to solve?

It’s often hard to find when ancient problems were first posed. Consider trisecting the angle, one of the other classic Greek geometry challenges. Trisecting the angle was proved impossible in 1836 or 1837 by Wantzel. So, it would have to have been posed by about 461 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^p−1(2^p − 1) is an even perfect number whenever 2^{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 – 1 = 2319 years!

Can you help me out here?