In the News: No Prizes for Guessing

Poincaré conjecture


What is it?

"If we stretch a rubber band around the surface of an apple, we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. If we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut." [Clay Mathematics Institute]

What do I get for solving it?

This is the conjecture that Perelman is thought to have cracked - they're still checking his workings - so you'd better get a move on if you want to claim the $1m from the Clay Institute in Boston.

Fermat's Last Theorem


What is it?

"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvellous demonstration of this proposition that this margin is too narrow to contain." [As scribbled by Fermat in the margin of an ancient Greek text, Arithmetica]

What do I get?

Unfortunately, Professor Andrew Wiles of Princeton University has already snaffled the £30,000 prize set up by Paul Wolfskehl. But given that the tools Wiles used to prove it in the 1990s didn't exist when Fermat claimed to have solved it, the search is still on for a simpler solution. Worth a go.

Riemann hypothesis


What is it?

"The distribution of prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician GFB Riemann observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function called the Riemann Zeta function. This has been checked for the first 1,500,000,000 solutions." [Clay Institute]

What do I get?

It's another of those $1m questions. If you can get it to 1,500,000,001, why not email prize.problems@claymath.org


Goldbach's conjecture


What is it?

"It seems that every number that is greater than 2 is the sum of three primes." 1 is allowed to count as a prime number, even though, strictly speaking, it's not.

What do I get?

Publisher Faber & Faber offered $1m to anyone who could prove it, to promote Uncle Petros and Goldbach's Conjecture, by the Greek author Apostolos Doxiadis. They set a two-year time limit, which expired on March 20 2002, but you could always try the held-up-in-the-post trick.

Collatz problem


What is it?

Take any positive whole number. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat this process over and over again. Eventually you will reach 1. It's true for every number tested so far.

What do I get?

Professor Bryan Thwaites has offered £1,000. All sensible mathematicians are waiting till it's upped to a million.

Happy ending problem


What is it?

"The problem of determining for the smallest number of points in general position in the plane (ie no three of which are collinear), such that every possible arrangement of points will always contain at least one set of points that are the vertices of a convex polygon of sides." [Wolfram Mathworld] No, us neither.

What do I get?

No monetary reward here, but the problem got its name because two of the researchers working on it ended up married. Surely that's prize enough?

The Euler-Mascheroni constant


What is it?

The Euler-Mascheroni constant y is also known as the Euler constant, but should not be confused with the constant e = 2.718281 ... You want more, look it up.

What do I get?

Mathematician GH Hardy offered his Savilian professorship at Oxford to anyone who could prove that y was irrational, though given that this was in the 1920s, the present incumbent might have something to say about that.