Monday, 14 December 2015

marginalia or many pleasant facts about the square of the hypotenuse

Having taken more than three hundred fifty years to prove since the claim was first coyly presented and in fiction and popular culture, the final, mysterious conjecture by poly-math and number theorist Pierre de Fermat probably did not strike the mathematician himself nor its originally prompter as particularly significant. Fermat’s Last Theorem, as it has come to be known, was inspired by a book of lemmas by an ancient Greek mathematician called Diophantus of Alexandria. For this scholar, considered the father of algebra (not a terror-organisation and ought not to be intimidating to the public like one) for inventing variable notation and despite his monumentally new paradigm of recognising fractions as legitimate numbers, Diophantus (at least in his surviving books) did not break with the traditional penchant for finding whole number solutions for problems.
Finding a nice round solution is much more satisfying and resonates far more—I think, even given computational power that masks the ugly, irrational bits. The book of Diophantus that Fermat was reading, the Arithmetica, was rather a conversational, speculative investigation that proffered that right angled triangles (following the Pythagorean Theorem, a² + b² = c²) exist where the sides of the triangle work out to be whole numbers: 3² + 4² = 5² or 9 + 16 = 25. There seemed to exist as many solutions, however, where the answers were not so tidy. Seeing this, Fermat wondered if the application could be expounded to higher exponents (and thus dimensions—something squared is a flat surface as opposed to a three-dimensional cube) and running with it, asserted that no whole number solutions can exist for a³ + b³ = c³ or higher powers up to infinity. This assertion was scribbled, coyly, in the margin of Diophantus’ ponderings with the aside that there’s a nifty proof for this necessity but not enough room to write it here. Perhaps Fermat felt that the problem was not so pressing and never again returned to that particular problem, leaving generations to wrestle with it after his notes were discovered. There’s a whole cosmos of unsolved equations that might pose more appreciable and immediate significance if explained, and while there’s no obvious application in understanding why what Fermat declared is ultimately true, the insight and techniques developed in trying to find the answer have propelled mathematics forward and have enabled all sorts of progress in understanding and has shaped the modern world. I can’t claim any understanding of the famous proof and my brain starts to hurt from it, but I wonder if it also shows, for this celebrated and veteran conundrum, why it’s the case—that whole numbers are not transcendent.