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Although the proof of Dirichlet's Theorem makes use of calculus and analytic number theory, some proofs of examples are much more straightforward. In particular, the proof of the example of infinitely many primes of the form 4 n + 3 {\displaystyle 4n+3} makes an argument similar to the one made in the proof of Euclid's theorem (Silverman 2013).
This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later. [2]
It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. [ 1 ] [ 2 ] It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function ) and additive number theory (such ...
Dirichlet's theorem may refer to any of several mathematical theorems due to Peter Gustav Lejeune Dirichlet. Dirichlet's theorem on arithmetic progressions;
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. [1] It determines the rank of the group of units in the ring O K of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.
Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. [15] He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem.
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 ...
The Vorlesungen contains two key results in number theory which were first proved by Dirichlet. The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These ...