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In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that < <.First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan.
See proof of Bertrand's postulate for the details. [ 9 ] Erdős proved in 1934 that for any positive integer k , there is a natural number N such that for all n > N , there are at least k primes between n and 2 n .
Bertrand's postulate; Proof of Bertrand's postulate; Bonse's inequality; Brun–Titchmarsh theorem; Brun's theorem; C. Chen's theorem; D.
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational
In November of the same year, Larsen published a paper titled "Bertrand's Postulate for Carmichael Numbers" [9] on the open access repository arXiv that made a more consolidated proof of Maynard and Tao's postulate but involving Carmichael numbers into the twin primes conjecture and attempting to shorten the distance between the numbers per ...
Proof of Bertrand's postulate; Fermat's theorem on sums of two squares; Two proofs of the Law of quadratic reciprocity; Proof of Wedderburn's little theorem asserting that every finite division ring is a field; Four proofs of the Basel problem; Proof that e is irrational (also showing the irrationality of certain related numbers) Hilbert's ...
Bertrand's postulate. Proof of Bertrand's postulate; Proof that the sum of the reciprocals of the primes diverges; Cramér's conjecture; Riemann hypothesis. Critical line theorem; Hilbert–Pólya conjecture; Generalized Riemann hypothesis; Mertens function, Mertens conjecture, Meissel–Mertens constant; De Bruijn–Newman constant; Dirichlet ...
In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. [1] At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: