Legendre

Legendre is a surname most often associated with Adrien-Marie Legendre (1752–1833), a French mathematician and astronomer. Legendre made significant contributions to number theory, analysis, and celestial mechanics, and he held professorships alongside membership in the French Academy of Sciences. He helped develop the method of least squares and contributed to the theory of astronomical computations.

In mathematics and its applications, several results bear his name: Legendre polynomials P_n(x) and the associated Legendre functions arise from Legendre's differential equation and form an orthogonal basis on [-1,1], widely used in physics, numerical analysis, and spherical harmonics. Legendre transformation, a fundamental tool in convex analysis, relates a function to its Legendre transform, providing a bridge between Lagrangian and Hamiltonian formalisms in mechanics. In number theory, the Legendre symbol (a/p) encodes quadratic residue information modulo an odd prime p, and Legendre's three-square theorem characterizes integers that are expressible as sums of three squares. Other Legendre constructs include Legendre's formula for the exponent of a prime p in the factorial n!, Legendre's conjecture on the existence of a prime between n^2 and (n+1)^2, and Legendre's contributions to the theory of congruences.