Fermat's Last Theorem

It fascinates me that some problems sound as though they must have complicated solutions but turn out to have relatively simple ones and vice versa. Concepts in science that can be defined mathematically often have an elegance that appeals on an aesthetic level. Consider, for example, all of the physical processes that can be described by neat differential equations such as the wave equation. We take it for granted that we can decompose a fluctuation over time into its frequency components (everyone basically knows what a frequency spectrum is), but beneath this everyday concept lies a compact elegance in actually being able to describe almost any waveform as a set of simple oscillations.

I think most non-mathematicians like myself find it remarkably strange that some simple-looking problems in mathematics require such appallingly complicated approaches to solve them. Here’s a classic example - the infamous last theorem of Fermat’s which remained unproved for over 350 years:

xⁿ + yⁿ = zⁿ (n>2) has no solution for x,y,z whole numbers not equal to 0.

How can the only known solution (Andrew Wiles 1995) require more than a hundred pages of sophisticated mathematical logic built on several centuries of earlier mathematical discoveries? Maybe part of it is that playing with whole numbers sounds like child’s play. Of course the truth is quite the opposite – if you allow non-whole-number x, y and z there are infinitely more solutions than children with nth root buttons on their calculators. Now in fact you could ask a fairly young child who understands exactly what a cube is and can count to play with building blocks and look for solutions corresponding to n=3. You would just get them to make up two cubes and then use exactly those blocks to try and make a third cube. But Euler discovered in 1753 that there were no such solutions and the two-page proof is nicely set out in Larry Freeman's blog.

On the subject of simplicity, Euler’s identity (e^iπ = -1) has to be one of the most aesthetically delightful things that one can scribble down on a piece of paper! It takes two of mathematics’ most fundamental constants (e and π) - both are irrational numbers needing an infinite number of decimal places to fix them “precisely” on a ruler – and combines them with the so-called imaginary number i=√(-1) to end up with a number as simple as -1. For anyone not familiar with the square root of a negative number, if your ruler represents the usual line where “regular” numbers are defined, the number i actually lives perpendicular to your ruler to form the so-called complex plane. This turns out to be great for describing circular motion. Just think of one of the hands of a clock rotating around - the clock-face is the complex plane. Perhaps even more amazing on the simplicity front is that one doesn’t need to dream up numbers more complex than “complex” numbers to do the most advanced math in the world.

Which brings us back to Wiles. The complicated nature of his proof renders it basically incomprehensible to any but the most skilled professional mathematicians. Yet elegance and simplicity are relative attributes. The proof rests on unifying two branches of mathematics that had long been suspected to be equivalent. And in fact the final proof became simpler than the initial flawed proof that he presented in 1993. Fermat himself claimed that he had a proof of his conjecture. If so, this must have been considerably simpler than Wiles’, given the level of mathematical sophistication available in Fermat’s day. Fermat was a very able mathematician, but he tended to give only the main outline of a proof which implied rather than showed unequivocally that he had a rigorous proof. His last theorem has always been a tease for mathematicians, because his claim of a proof is quite unsubstantiated and was only discovered posthumously in the margins of one of his books.

In his highly entertaining book, Fermat’s Enigma, Simon Singh recounts all the drama associated with the development of mathematics from Pythagoras to the present day, and the various attempts along the way to prove this particularly stubborn mathematical riddle. He relates the ordeal of the 20-year old French mathematician Evariste Galois, who stayed up the night before a duel (which he was certain to lose) to record for posterity his major mathematical insights which had been lost by the organizers of a mathematics competition. Fortunately Wiles didn’t have quite the same time pressure to revise his flawed initial proof. Instead, he retreated into what must have nevertheless been a private nightmare for over a year, and was able, with the help of a former student of his, to forge another – this time faultless - logical passage through the arcane mathematical landscape of cyclotomic number fields to connect at least some elliptic curves to modular forms – enough to prove the last theorem once and for all.