### 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 Exilewhich 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 Birdswritten 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 $e^x$ is transcendental for every nonzero algebraic number $x$. Taking $x = i \pi$ this implies that $\pi$ 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 $2^{p−1}(2^p − 1)$ 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 = 2320 years!

Can you help me out here?

## 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.