Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory; Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in statistics and number theory completed that of Legendre.
In 1830 he gave a proof of Fermat's last theorem for exponent n = 5, which was given almost simultaneously by Dirichlet in 1828.
In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss. He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1796 conjecture of the Prime number theorem was rigorously proved by Hadamard[?] and de la Vallée Poussin in 1898.
Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.
In theoretical mechanics, he is known for the Legendre transform[?], which is used to go from the Lagrangian to the Hamiltonian formulation of mechanics.
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