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 eix 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 2p−1(2p − 1) is an even perfect number whenever 2p − 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?

18 Responses to The Hardest Math Problem

  1. jb says:

    The theorem you attribute to Euclid is true and was no doubt known to him, but it probably isn’t what you meant: 2^{p-1} is rarely prime!

  2. artusartos says:

    Puzzle. Why did I subtract 1 here? The question was not posed in the beginning of the year or not settled at the end the year?

    I’m somewhat confused with the last problem. How can 2^(p-1) be prime when it’s divisible by 2?

    • John Baez says:

      Artusartos wrote:

      Puzzle. Why did I subtract 1 here? The question was not posed in the beginning of the year or not settled at the end the year?

      Actually I did it because there was no “year zero”. But yes: if someone formulated a math problem at the very end of 1 BC and someone solved it at the very start of 1 AD, it would have been open for just one day!

      I’m somewhat confused with the last problem. How can 2^(p-1) be prime when it’s divisible by 2?

      That was a typo, which I’ve now fixed.

  3. Ishi Crew says: and they have one for 2295. both of these have interesting factorizations into primes –eg involve a 17 and a 41 this probably explains why it took that long.

    it you believe in the block universe theory (possibly einstein endorsed that) the hardest math problem to solve (in terms of years it took to solve) might be either reimann conjecture , or p=/np? , or 3n+1 or collatz problem.

    If you are eternalist (belive in block universe). then one could look up on the eternalist calendar to see the date when they were solved.

    perhaps maybe in 1729^2 AD. .

    hence all math problems have been solved. some of the solutions are at the end of the book —eg on pg 1729^2. In future editions they will all be covered in 1st 5 pages (using lebesque’s/vitali/bellman universal covering’).

    my area is sort of on total lockdown for coronoavirus. but not block down. (though a few people around here did get gunned down recently which is a sort of ‘community concern’.) one can still go outside to the park— saw a muskrat swimming in the creek, some trees recently cut down by beavers, heard frogs in the swamp, and more.)

  4. Bob says:

    Philoponus tells of how the Athenians, in 432 B.C, suffering from the plague, consulted Plato in regards to the Delian problem of duplicating the area of a cube; later proven not constructible by Pierre Wantzel in 1837[1].

               -To think that two and two are four
                    and neither five nor three,
                 The heart of man has long been sore
                    And long 'tis like to be.
                         - A. E. Housman [2]

    [1] “The Mathematical Puzzles of Sam Lloyd”, Martin Gardner,1950
    [2] “Fantasia Mathematica”, Clifton Fadiman,1958

  5. The 1 that you subtracted is the missing year zero.

  6. Arch1 says:

    It is interesting that it took a Gauss to see the connection between this problem in geometry and the roots of certain polynomials.

  7. Blue Minnow says:

    I would say that factorisation – in particular, efficient factorisation of arbitrarily large integers – is the oldest unsolved mathematical problem known to our particular variety of sapient species.

    In terms of age, a quick wikipedia lookup suggests that the Egyptians understood about the existence of prime numbers circa 1550 BC (of which we have historical evidence via the Rhind Mathematical Papyrus). It is a small inference to posit that they understood about the difficulty of factoring, too, though they evidently would not have been interested in numbers much larger than several tens of thousands.

    So that would make the above problem upwards of 3500 years old, and possibly closer to 4000 years old.

  8. A guess a contender for longest problem to solve in math would be the parallel postulate. It seems like the timeline is sketchier here than for squaring the circle, but starts also starts with the ancient Greeks and is resolved by 19th century Europeans.

    I don’t know of the earliest known account of the problem, but it had to be known to whoever wrote the 5 axioms in Elements that the fifth one stands out. From a little digging it seems that book 1, which contains the axioms, was thought to mostly come from Pythagoras (c. 570–495 BC) . However, while Pythagoras (or his school) provided the proofs it’s not clear to me that anyone before Euclid explicitly stated the axioms as assumptions.

    Also unclear to me is how would one consider it solved? Yes, Gauss and others worked on non-Eulcidean geometry but as far as I can tell they didn’t prove anything about the equiconsistency between non-Euclidean and Eulcidean geometries. The earliest proof of that I can find was by Beltrami in 1868.

    • John Baez says:

      That’s an interesting contender. Thanks! I hadn’t thought of ‘proving the parallel postulate from other axioms’ as a problem the Greeks contended with… but it probably was, at some point. Didn’t Euclid introduce that postulate after the rest, in his Elements?

  9. The hardest applied math problem might be Navier-Stokes. Since COVID-19 may get me any day now, here’s a practical analytic solution that I can leave for future generations to validate ;)

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