# Playing the Numbers Game

A mathematician has informed me that any prime number that is divisible by 4 with remainder 1 is always the sum of two squares.  Try it out: 13 divided by 4 is 3 remainder 1, and is the sum of 9 and 4.  17 divided by 4 is 4 remainder 1, and is the sum of 16 and 1.  29 divided by 4 is 7 remainder 1, and is the sum of 25 and 4.  37 divided by 4 is 9 remainder 1, and is the sum of 36 and 1.  41 divided by 4 is 10 remainder 1, and is the sum of 25 and 16.  53… 49 and 4.  61… 36 and 25.  And so on.

I had three questions for my mathematician.  First, is there a proof, or do we just have to examine each prime number sequentially, as above, and keep going ad infinitum, looking for an exception?  I suppose if there is a proof, then somebody of sufficient brain power would merely shrug and say, “Well, it’s a statement of the bleedin’ obvious, in’ it, mate.”

Second, are there always only two unique squares to make the addition, or might two different squares add up to the same prime?  My gut instinct leans to the former, but who knows?

Lastly, how do people get into this stuff?  Why would you elect to track this elusive spoor?  I mean, why on earth would you go to the bother of dividing a prime number by 4, and then searching for squares that add up to it?  What on earth is the point?  I suppose people experience a sense of delight, and perhaps even awe, when they uncover some strange and unlikely numerical relationship, which seems to tantalise us with a hint of the reasoned governance of the universe.  A number theorist who stumbles upon a proof hitherto unknown, conscious of being the only consciousness in the cosmos apprised of such knowledge, must feel quite Olympian.  You might suppose that somebody possessed of such knowledge, uniquely privileged to be privy to arcane truths, might be in a rush to publish, to establish prestige, precedence, and a reputation.  Yet, for a mathematician, it might be that there is a seductive attraction to keeping a secret.  This knowledge, and its power, is mine and mine alone.  Thus, Newton kept the calculus to himself, the way somebody who purloins a Stradivarius from a railway compartment, or a Vermeer from an art gallery, can only admire the prize, alone, behind closed doors.