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An analogue of Dirichlet's theorem holds in the framework of dynamical systems (T. Sunada and A. Katsuda, 1990). Shiu showed that any arithmetic progression satisfying the hypothesis of Dirichlet's theorem will in fact contain arbitrarily long runs of consecutive prime numbers. [7]
In 1837, Dirichlet proved his theorem on arithmetic progressions using concepts from mathematical analysis to tackle an algebraic problem, thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions.
In 1837 he published Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions.
Dirichlet's theorem may refer to any of several mathematical theorems due to Peter Gustav Lejeune Dirichlet. Dirichlet's theorem on arithmetic progressions; Dirichlet's approximation theorem; Dirichlet's unit theorem; Dirichlet conditions; Dirichlet boundary condition; Dirichlet's principle; Pigeonhole principle, sometimes also called Dirichlet ...
Dirichlet's theorem on arithmetic progressions. In 1808 Legendre published an attempt at a proof of Dirichlet's theorem, but as Dupré pointed out in 1859 one of the lemmas used by Legendre is false. Dirichlet gave a complete proof in 1837. The proofs of the Kronecker–Weber theorem by Kronecker (1853) and Weber (1886) both had gaps. The first ...
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
Dirichlet's approximation theorem (Diophantine approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Equidistribution theorem (ergodic theory) Erdős–Kac theorem (number theory) Euclid's theorem (number theory) Euclid–Euler theorem (number theory) Euler's theorem ...
See also Dirichlet's theorem on arithmetic progressions. As of 2020, the longest known arithmetic progression of primes has length 27: [4] 224584605939537911 + 81292139·23#·n, for n = 0 to 26. (23# = 223092870) As of 2011, the longest known arithmetic progression of consecutive primes has length 10. It was found in 1998.