Sárközy's theorem

Sárközy's theorem states that in any sufficiently large set of whole numbers, you'll find two numbers whose difference is a perfect square (like 1, 4, 9, etc.). Specifically, if the set has a positive density—meaning it contains a good proportion of numbers—it will always contain pairs that differ by a perfect square, no matter which large set you choose. This result highlights a fundamental pattern: large enough sets inevitably include pairs separated by perfect squares, revealing an intrinsic structure within the integers.