Sophie Germain by Dora Musielak

Author:Dora Musielak
Language: eng
Format: epub
ISBN: 9783030383756
Publisher: Springer International Publishing

What Fermat meant is that if is a positive integer , the equation has no non-zero integer solutions for , and , i.e., no three integers , and exist, such that , which satisfy the equation.

Fermat’s declaration was rather intriguing. It was known as a theorem because Fermat claimed to have proved it. However, no proof was ever found.

Euler and Fermat’s Theorems

Early in the eighteenth century, number theory was still regarded as a minor subject, largely of recreational interest. Mathematicians were more interested in applying the new infinitesimal calculus invented by Newton and Leibniz to solve a variety of problems in physics and astronomy.

After Fermat, Leonhard Euler gave luster and depth to number theory, ushering it into the mainstream of mathematics. Euler was the most prolific mathematician ever—and one of the most influential—and when he turned his attention to number theory, the subject attained a higher status. Combining pure genius, a superhuman ability to perform complex mathematical analysis and computations in his head, and enormous intellectual sensibility, Euler was precisely the right mathematician to deal with the unproven assertions of Fermat.

It was Christian Goldbach who (in a letter dated December 1729) introduced the twenty-two-year old Euler to the work of Fermat. In the postscript of that letter, Goldbach wrote:P.S. Notane Tibi est Fermatii observatio omnes numeros hujus formulae nempe 3, 5, 17, etc. esse primos, quam tamen ipse fatebatur sc demonstrare non posse, et post eum nemo, quod sciam, demonstravit. [P.S. Note Fermat’s observation that all numbers of this form , that is 3, 5, 17, etc. are primes, but he confessed that he did not have a proof, and after him, no one, to my knowledge, has proved it.]10

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