A vignette of Number theory for RMO — Sums of Higher Powers and Fermat’s Last Theorem

In a previous blog, we discovered that the equation a^{2}+b^{2}=c^{2}  has lots of solutions in whole numbers a, b and c. it is natural to ask whether there are solutions  when the exponent 2 is replaced by a higher power. For example, do the equations

a^{3}+b^{3}=c^{3} and a^{4}+b^{4}=c^{4} and a^{5}+b^{5}=c^{5} have solutions in nonzero integers a, b and c? The answer is NO. Sometime around 1637, Pierre de Fermat showed that there is no solution for exponent 4. During the 18th and 19th centuries, Karl Friedrich Gauss and Leonhard Euler showed that there is no solution for exponent 3 and Lejeune Dirichlet and Adrien Legendre dealt with exponent 5. The general problem of showing that the equation a^{n}+b^{n}=c^{n} has no solutions in positive integers if n \geq 3 is known as “Fermat’s Last Theorem”. It has attained almost cult status in the 350 years since Fermat scribbled the following assertion in the margin of  one of his books:

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or  in general any higher power than the second into powers of like degree. I have discovered a truly remarkable proof which this margin is too small to contain.

Few mathematicians today believe that Fermat had a valid proof of his “Theorem,” which is called his Last Theorem because it was the last of his assertions that remained approved. The history of Fermat’s Last Theorem is fascinating, with literally hundreds of mathematicians making important contributions. Even a brief summary could easily fill a book. This is not our intent in this blog, so we will be content with a few brief remarks.

One of the first general results on Fermat’s Last Theorem, as opposed to verification for specific exponents n, was given by Sophie Germain in 1823. She proved that if both p and 2p+1 are primes then the equation a^{p}+b^{p}=c^{p} has no solutions in integers a, b and c with p not  dividing the product abc. A later result of a similar nature, due to A. Wieterich in 1909, is that the same conclusion is true if the quantity 2^{p}-2 is not divisible by p^{2}. Meanwhile, during the latter part of the 19th century a number of mathematicians, including Richard Dedekind, Leopold Kronecker, and especially Ernst Kummer, developed a new field of mathematics called algebraic number theory, and used their theory to prove Fermat’s Last Theorem for many exponents, although still a finite list. Then, in 1985, L. M. Adleman, D. R. Heath-Brown, and E. Fouvry used a refinement of Germain’s criterion together with difficult analytic estimates to  prove that there are infinitely many primes p such that a^{p}+b^{p}=c^{p} has no solutions with p not dividing abc.

Sophie  Germain (1776-1831) Sophie Germain was a French mathematician who did important work on number theory and differential equations. She is best known for her work on Fermat’s Last Theore, where she gave a simple criterion that suffices to show that the equation a^{p}+b^{p}=c^{p} has no solutions with abc not divisible by p. She also did work on acoustics and elasticity, especially the theory of vibrating plates. As a mathematics student, she was forced to take correspondence courses from Ecole Polytechnique in Paris, since they did not accept women as students. For a similar reason, she began her extensive correspondence with Gauss using the pseudonym Monsieur Le Blanc, but when she eventually revealed her identity, Gauss was delighted and sufficiently impressed with her work to recommend her for an honorary degree at the University of Gottingen. 

In 1986, Gerhard Frey suggested a new line of attack on Fermat’s problem using a notion called modularity. Frey’s idea was refined by Jean-Pierre Serre, and Ken Ribet subsequently proved that if the Modularity Conjecture is true, then Fermat’s Last Theorem is true. Precisely, Ribet proved that if every semistable elliptic curve is modular then Fermat’s Last Theorem is true. The Modularity Conjecture which asserts that every rational elliptic curve is modular, was at that time a conjecture originally formulated by Goro Shimura and Yutaka Taniyama. Finally, in 1994, Andrew Wiles announced a proof that every semistable rational elliptic curve is modular, thereby completing a proof of Fermat’s 350 year old conundrum. Wiles’s proof, which is a tour de force using the vast machinery of  number theory and algebraic geometry, is far too complicated for us to describe in detail, but we will try to convey the flavour of his proof much later in these blogs. 

Few mathematical or scientific discoveries arise in vacuum. Even Sir Isaac Newton, the transcendent genius not noted for his modesty, wrote that “If I have seen further than others, it is by standing on the shoulders of giants.” There are several mathematicians before Sir Andrew Wiles, whose work directly or indirectly contributed to the final proof of Fermat’s Last Theorem.

More later…

Nalin Pithwa

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