In the first few lectures of Graph Theory, the lecturer (Paul Russell) presented a cute application of Ramsey theory to Fermat’s Last Theorem. It makes a great introduction to the general process of casting a problem in one branch of mathematics as a problem in another and is the perfect size for a blog post, so I thought I’d talk about it.
The setup is as follows. One naive way to go about proving the nonexistence of nontrivial integer solutions to (that is, solutions such that are not equal to zero) is using modular arithmetic; that is, one might hope that for every it might be possible to find a modulus such that the equation has no nontrivial solution . To simplify matters, we’ll assume that are relatively prime to , or else there is some subtlety in the definition of “nontrivial” (e.g. we might have not divisible by but .) Note that it might be the case that is not relatively prime to a particular nontrivial solution in the integers, but if we can prove non-existence of nontrivial solutions for infinitely many (in particular, such that any integer is relatively prime to at least one such ) then we can conclude that no nontrivial integer solutions exist.
By the Chinese Remainder Theorem, this is possible if and only if it is possible with a prime power, say . If is relatively prime to , this is possible if and only if it is possible with . This is because given a nontrivial solution we can use Hensel’s lemma to lift it to a nontrivial solution for any (and even to ), and the converse is obvious. (Again to simplify matters, we’ll ignore the finitely many primes that divide .) So we are led to the following question.
For a fixed positive integer do there exist infinitely many primes relatively prime to such that has no nontrivial solutions?
As it turns out, the answer is no. In 1916 Schur found a clever way to prove this by proving the following theorem.
Theorem: For every positive integer there exists a positive integer such that if is partitioned into disjoint subsets , then there exists such that there exist with . In other words, the Schur number exists. (Note that I am using a convention which is off by .)
If we let be a prime greater than and let the be the cosets of the subgroup of powers in , which has index at most , we obtain the following as a corollary.
Corollary: Fix a positive integer . For any sufficiently large prime , there exists a nontrivial solution to .
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