If x is an even integer that can be written as the sum of two perfect squares, prove that x / 2 can also be written as the sum of two perfect squares.


Solution

Since x can be written as the sum of two perfect squares, x = a2 + b2.

Since x is even, a2 and b2 are either both odd or both even (i.e., have the same parity).

Since parity does not change when you square a number, that means a and b also have the same parity.

Now x / 2 = (a2 + b2) / 2 = ((a + b) / 2)2 + ((a – b) / 2)2.

Since a and b are the same parity, that means (a + b) / 2 and (a – b) / 2 are both integers. Therefore x / 2 can be written as the sum of two perfect squares.

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