How far apart can squares be?

Consider the squares: 0, 1, 4, 9, 16, 25, 36…

“No two squares are 6 apart,” I say. After some subtractions you believe me.

Exercise: prove no two squares are 6 apart.

“Nor are any two squares 134 apart,” I say. You look at me in surprise. After some puzzlement, inspiration strikes.

Exercise: prove that any two squares differ either by an odd number or by multiple of 4.

“In fact,” I say, “the set {2, 6, 10, 14, …} = {4k + 2} completely describes how far apart two squares cannot be…”

Exercise: given any n not of the form 4k + 2, prove that it is possible to find two squares that are n apart.

“… with one exception.”

Exercise: find the exception. Find what was wrong with your previous proof. Convince yourself the proof is now correct and there are no other exceptions.

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