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, …} = {4*k* + 2} completely describes how far apart two squares cannot be…”

*Exercise: *given any *n* not of the form 4*k* + 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.