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In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also investigated the primality of the Fermat numbers 22n+1{\displaystyle 2^{2^{n}}+1}, and Marin Mersenne studied the Mersenne primes, prime numbers of the form 2p−1{\displaystyle 2^{p}-1} with p{\displaystyle p} itself a prime. Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes 12+13+15+17+111+⋯{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots }. At the start of the 19th century, Legendre and Gauss conjectured that as x{\displaystyle x} tends to infinity, the number of primes up to x{\displaystyle x} is asymptotic to x/log⁡x{\displaystyle x/\log x}, where log⁡x{\displaystyle \log x} is the natural logarithm of x{\displaystyle x}. Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving this. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.

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