Fermats

Fermats refers to the body of work and mathematical objects associated with Pierre de Fermat (1607–1665), a French lawyer who became a leading figure in early modern mathematics. Although he pursued law professionally, he conducted extensive investigations in number theory, geometry, probability, and optics, often communicating insights in letters and marginal notes. The name Fermats is thus attached to a number of theorems, constructions, and sequences that continue to influence modern mathematics.

Fermat's Last Theorem states that there are no positive integers a, b, c and integer n greater than 2 satisfying a^n + b^n = c^n. He stated the claim in 1637 in the margin of Diophantus's Arithmetica, but a complete proof was not provided until Andrew Wiles published one in 1994. Fermat's Little Theorem concerns modular arithmetic: if p is a prime and a is not divisible by p, then a^(p−1) ≡ 1 mod p; this result remains a foundational tool in number theory. In optics, Fermat's principle posits that light travels along paths that extremize travel time, a concept foundational to the study of refraction and reflection. The Fermat numbers, defined by F_n = 2^(2^n) + 1, form a sequence that was once believed to yield many primes; the first five terms are prime, but F_5 and later terms are composite.

The Fermat point is a construction in triangle geometry that minimizes the sum of distances from a point to the triangle's vertices. Fermat's influence spans number theory, geometry, and mathematical analysis, and his methods, including the technique of infinite descent, shaped subsequent developments in mathematics.