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In number theory, Bertrand's postulate is the theorem that for any integer >, there exists at least one prime number with n < p < 2 n − 2. {\displaystyle n<p<2n-2.} A less restrictive formulation is: for every n > 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that
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.
This striking formula is one of the so-called explicit formulas of number theory, and is already suggestive of the result we wish to prove, since the term x (claimed to be the correct asymptotic order of ψ(x)) appears on the right-hand side, followed by (presumably) lower-order asymptotic terms.
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 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
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 ...
Bertrand's postulate; Proof of Bertrand's postulate; Bi-twin chain; C. Chebyshev's bias; Closing the Gap: The Quest to Understand Prime Numbers; ... Formula for primes;
Another proof that the harmonic numbers are not integers observes that the denominator of must be divisible by all prime numbers greater than / and less than or equal to , and uses Bertrand's postulate to prove that this set of primes is non-empty.