The theorem itself isn't that hard to understand, you just need the definitions of some terms. The density of a set S of natural numbers (1, 2, 3, ...) less than N is | S |/N (e.g. the probability that a random number less than N is in S). The theorem says that for any density other than 0, there is a number N such that any subset of {1, 2, ..., N} with the chosen density contains a pair of numbers that differ by a perfect square. That probably reads a little awkwardly, but you'll get very familiar with statements of that form when you take calculus.

The proof is hard, though it's a bit easier than other proofs of the same theorem. One of the things that makes number theory interesting is that theorems that are very easy to state can be very hard to prove.

This kind of thing will probably be impenetrable for a while, but if you're at all interested in number theory there's a cheap Dover reprint of Underwood Dudley's Elementary Number Theory that doesn't require much more than high school algebra.

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