In a letter to Leonhard Euler, dated 7 June 1742, Christian Goldbach, a not especially eminent mathematician, speculated that every even number greater than 2 was the sum of at least one pair of primes (a prime being an integer divisible solely by itself and 1). Thus 10 = 7+3, 32 = 13+19, 84 = 67+17, ad infinitum. To the great Euler, it probably seemed a trifling puzzle not worth racking his brain over.
To this day, however, and despite its undoubted truth - it has been tested literally billions of times with no counterexample ever found - Goldbach's Conjecture has defied all attempts at proof.
If it were ever conclusively proved (mathematicians have got awfully close), the Conjecture's impact on the current state of number theory would be minimal. Yet, because of its inexplicable recalcitrance to emerge into the light of reason, its vanquisher would achieve instant celebrity, just as Goldbach's own name has survived exclusively through his having posed the problem.
Which means that, if no professional mathematician any longer deigns to expend the necessary time, trouble and nervous tension trying to prove it, the Conjecture (like Fermat's Last Theorem) has always been a favourite of gifted obsessives.
Just such an obsessive is the Uncle Petros of Apostolos Doxiadis's short novel (long novella, rather). Petros is a crochety, eccentrically antisocial codger, a reclusive skeleton in his family's closet, living alone and virtually friendless on the periphery of the international mathematical community of which he might have been one of the glories. His tragedy is that he has wasted - and eventually comes to know he has wasted - what were potentially the best and most fruitful years of his career (mathematicians are notoriously premature burn-outs) poring over a problem that is simply too difficult. It is, moreover, a problem whose sole interest, given that it has no applicability in the real world, is its own solution.
Refusing to publish interim results, often the most useful part of any quixotic mathematical challenge, for fear that some imagined rivals might steal a march on him (precisely the paranoid attitude adopted by Andrew Wiles, who finally cracked Fermat's Last Theorem), he has to stand on the sidelines, unacclaimed, while these results are attributed to other, less fastidious mathematicians who have arrived at them independently and have absolutely no reason for not at once going public.
Petros is obliged to admit defeat at last when, in the early Thirties, the logician Kurt GÃ¶del (a historical figure, of course, who makes a brief cameo appearance) publishes his two shattering Theorems of Incompleteness. They demonstrate that there exist mathematical statements that, even if true, can never be proved true - although, alas, there is no method of knowing in advance which they might be.
Because GÃ¶del's achievement is as crucial to the understanding of Doxiadis's novel as it is to the history of contemporary ideas, it's important to get it right. Thus its implication is not that mathematicians don't yet, but one day might, possess the intellectual tools that would permit them to answer every mathematical question. GÃ¶del actually succeeded in proving that some of these questions would remain forever unprovable, or what is now called undecidable. And Goldbach's Conjecture, as Petros realises to his horror, could well be one of them.
That, naturally, is not the end of the story, since poor Petros is destined to rise again - but readers must discover for themselves what happens next. The plot even has a twist in the tail, although - in view of the fact that if Doxiadis, himself a mathematician, had indeed found a proof for the Conjecture, he would unquestionably have published it in a paper under his own soon-to-be-world-famous name instead of squandering it on a character in a work of fiction - that twist is just a tiny bit predictable.
Ultimately, though, this is not too serious a flaw, since the novel deserves to be read above all as a fascinating, extremely credible portrait of a tormented mindset - and as one of the very rare examples of what, after science fiction, might be termed maths fiction.
Gilbert Adair's novel, 'A Closed Book', is published by Faber