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.
