Fermat's Last Theorem/Gabriel Lamé and Augustin Louis Cauchy

Gabriel Lamé and Augustin Louis Cauchy

Sophie Germain’s results appeared to many to be the turning point, and that the definitive solution of the enigma was close. The French Academy of Science offered a series of prizes to whoever succeeded in proving Fermat’s last theorem. In that epoch the offer of prizes for the solution of mathematical enigmas was a common practice followed by many academies, given that one was thus able to direct the research of many scientists into specific areas of knowledge. These prizes were used whether for research into the solutions of practical problems or for the solution of theoretical problems such as Fermat’s theorem. The mathematicians Gabriel Lamé and Augustin Luis Cauchy announced being on the point of proving the theorem utilising similar techniques. These mathematicians set the Parisian salons on fire with statements and extracts of their respective proofs but without ever publishing the complete proofs. This situation dragged on for the whole of the month of April 1847 until the mathematician Joseph Liouville read to the academy of science a letter from the mathematician Ernst Eduard Kummer. Kummer analysing the few pieces of evidence on the proofs of Cauchy and Lamé had become aware that the proofs followed the same reasoning and that they contained a logical error that rendered them fallacious. Kummer had become aware that the two mathematicians had based their proof on a property known as unique factorisation. This stated in substance that every integer number could be broken down into a single series of prime numbers. Known since the times of Euclid (fourth century B.C.) it was used in very many proofs and is so important that the theorem that enunciated it is called the fundamental theorem of arithmetic. The problem was that the proof of Cauchy and Lamé utilised complex numbers and, in the field of complex numbers, this property is not necessarily true. The lack of unique factorisation could have been worked round with some techniques but however there remained some recalcitrant numbers. These numbers could in theory have been faced singly, but being infinite that would not have resolved the problem. Lamé became aware of the impossibility of finding a general solution, while Cauchy held his implementation not so dependent on factorisation and worked for several weeks on the subject before giving up. Finally, in 1856, seeing that no correct solution had been presented to the academy, it decided to withdraw the prize. It still in fact lacked 138 years to the final solution.