Unproved Mathematical Hypotheses

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The words conjecture and hypothesis indicate that these are not yet proved, although they appear to be true. Once the statement is proved it gets changed to being a theorem.

I'm not sure of the state of all these, and some of them may have been proved recently.

Many people, including bad amateur mathematicians, have the impression that there is huge money on offer for the proofs of these conjectures. Some of them have or had money prizes on offer, but this is the exception. In general, all you'll get for proving one of these is eternal gratitude and praise from the community of mathematicians.

The Collatz Conjecture

Take any positive integer. If it is odd, multiply it by 3 and add 1. If it is even, divide it by 2. Repeat indefinitely. You will eventually reach the cycle 4, 2, 1 repeated.

There was a £1,000 prize on offer for the proof of this. I don't know whether the prize still exists.

The Riemann Hypothesis

This is a complicated statement to do with the distribution of primes. The case for it being true is so strong that many other 'theorems' have been proved on the assumption that it is true - if it turns out to be false, these will all be false as well.

Goldbach's Conjecture

Every even number greater than 2 can be written as the sum of two primes.

The Odd Perfect Number Conjecture

There are no odd perfect numbers. That is, there are no odd numbers which are equal to the sum of all their divisors (including 1 but excluding the number itself).


Ones that were proved recently (in the last 50 years):

Densest Sphere Packing

The densest packing of identical spheres in space is the body centred cubic method. There are a number of variants of this which have the same density.

Proved by Hales, 2014

Fermat's Last Theorem

This one is a misnomer. Fermat claimed to have proved it, making it a theorem, but there's no evidence that he actually did. It seems likely that if he had a proof, it was a flawed one. Since the theorem has now been proved, it can be called a theorem but it was never Fermat's last one.

The expression a^n + b^n = c^n has no integer solutions for n > 2.

Proved by Andrew Wiles in 19951.

The Four Colour Map Theorem

Four colours are enough to colour the countries of any map on a plane so that no two countries that share a border are the same colour. This also applies to maps on a sphere or anything topologically equivalent, but not to more complex surfaces such as the surface of a torus.

Proved in 1976, although the proof used a computer to check lots of particular cases.

The Poincaré Theorem

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

This was proved in 2003, earning the mathematician a 1 million dollar prize.

1Wiles's original proof, presented in 1993, had an error in it.

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Infinite Improbability Drive

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