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Austria competition problems, 2010

Number theory Difficulty 4.5 AIME Prove it Austria

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

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.

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