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. 

I admire mathematicians, but I don’t envy them.  They always want to pursue a line of enquiry much further than the rest of us.  “And then you can generalise this result to n…” – that’s when I glaze over.  I have no desire to retire to my loft for two years to rediscover Fermat’s last theorem.  Some of the unsolved problems of number theory can be very simply stated, which is no doubt why professors of mathematics are bombarded by lay correspondents who think they’ve just made a big breakthrough on the back of an envelope.  A professor of mathematics at Glasgow only ever took on a couple of research postgraduates because he thought that unless you were world class you were wasting your time.  He was a Cambridge man, so I guess he might have sat at the feet of the great mathematician G. H. Hardy, who mentions this business of the primes, divided by 4 remainder 1 etc, in his A Mathematician’s Apology.  When I was reading English at Glasgow I applied for a scholarship to McGill University in Montreal and was up in front of a rather intimidating interview panel chaired by the professor of mathematics.  We got into a debate about the Cambridge literary critic F. R. Leavis.  I remember the prof of maths didn’t think much of Leavis, and I’m sure I said in a pompous undergraduate way, something like, “Oh I think you’ll find that Frank has a penetrating intellect.”  It so happened that my Senior Hons tutor in English was also a Cambridge man who had sat at the feet of Dr Leavis.  At the time I had no idea about the great rift between Leavis and another Cambridge man, C. P. Snow, who knew G. H. Hardy well and wrote a preface to his apologia.  So I was quite unconsciously walking across a mine field.

I didn’t get the scholarship to McGill. 

I guess professors of mathematics can be somewhat aloof.  There are two pieces of advice often given to youngsters, and they stand at opposite ends of a spectrum.  With the best will in the world you can blight somebody’s ambition by saying, “Don’t go down this path.  You’re not good enough.”  And I have even heard school teachers say to pupils, with much less goodwill, “You’ll never make anything of your life!”  Didn’t somebody say that to John Lennon?

The other end of the spectrum can be just as damaging.  “The world’s your oyster.  You can achieve anything, if you put your mind to it.”  But that need not necessarily be true.  I think the astronaut Tim Peake gave some very sound advice when Lauren Laverne cast him away on Desert Island Discs.  She asked him what advice he would give to somebody who wanted to pursue a goal which, at least statistically, he or she had little chance of achieving.  His advice was to go for it, and enjoy the journey.  If it doesn’t work out quite as you intended, still, something equally good will come along.  Wise words from a people person.  Chuck your hat into the ring and see what happens.  Much better than sitting in a garret solving the Riemann Hypothesis.                              

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