Fermat's Last Theorem/Sophie Germain
After Euler's progress, for around fifty years there were no improvements notwithstanding the fact that the last theorem had become the most famous problem in the theory of numbers. This situation changed radically thanks to Sophie Germain.
Sophie Germain was born in 1776 and died in 1831 and for all her life she had to fight against prejudice. In her society it was unthinkable that a lady of good standing dedicated herself to subjects such as mathematics, but Germain when small had read a book on the history of mathematics and remained fascinated by the death of Archimedes.
The legend recounts that when a roman soldier went to call Archimedes in order to conduct him in front of the centurion Archimedes refused to follow him being occupied with a geometric problem, for this reason the soldier ran him through. Germain was so struck by the fact that a man could have lost his life for mathematics that she decided to study it. Initially her decision was much opposed by her father, but with the passing of the years he had to resign himself to the wishes of his daughter deciding to support her. Germain found it very difficult to acquire modern mathematical techniques given that her tutors did not intend to teach them to her and she, being a woman, could not attend the university where courses in advanced mathematics were held. Germain then used the stratagem of passing herself off as a monsieur Le Blanc, a student who had withdrawn from the Ecole Polytecnique. Obviously she was not able to attend lessons but utilising this false identity she succeeded in getting herself the printed notes and the problems for the students attending that she solved and presented always under the same pseudonym. In the beginning the trick worked until the course professor, the great mathematician Joseph-Louis Lagrange, wanted to know the student who furnished those so ingenious solutions. Lagrange meeting Germain was surprised but pleased by the young woman and decided to help her in her study of the material.
Germain worked for years on the theory of numbers and also interested herself in Fermat's theorem. She obtained a result that she held to be very important but wanting confirmations on the validity of her discovery she decided to contact the then greatest authority, that is to say Carl Friedrich Gauss. Gauss had not interested himself in Fermat's theorem holding the enunciation in itself without interest, but when he received the letter from Germain he remained so impressed by her result as to dedicate himself to the problem and to confirm to Germain the validity of her method. Germain's idea was based on the use of a particular typology of prime numbers that were subsequently called Sophie Germain's prime numbers. For these prime numbers Germain succeeded in demonstrating that solutions of Fermat's theorem probably did not exist. Probably it meant that these eventual solutions would have had some properties so particular as to render the existence of these numbers difficult. Some of her colleagues analysing the problems defined by these prime numbers succeeded in proving that solutions did not exist for some of them, such as 5 or 7. Subsequently Gauss abandoned the theory of numbers in order to dedicate himself to applied mathematics and Germain without further support in the field of mathematics decided to concentrate on physics, where she made important contributions in the study of elastic vibrations.