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Problem 275

AMC 12 late, AIME early

Number theory Difficulty 4.5 Prove it Austria 2010 · Austria · 2010

Show that $2010$ cannot be written as the difference of two squares.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

We assume that there are two integers $x, y$ with
$2010 = x^2 - y^2 = (x - y)(x + y).$
The factors $(x - y)$ and $x + y = (x - y) + 2y$ have the same parity.

• If both of them were odd, the product $2010$ would be odd which also gives a contradiction.

• If both of them were even, the product $2010$ would be divisible by $4$ which gives a contradiction.

Therefore, there are no such numbers.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.