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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. [2]
Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane. Thus, as Gaussian primes extend over the plane and not only along a line, and doubling a complex number is not simply multiplying by 2 but doubling its norm (multiplying by 1+i), different ...
Bertrand's postulate; Proof of Bertrand's postulate; Bonse's inequality; Brun–Titchmarsh theorem; Brun's theorem; C. Chen's theorem; D. Dirichlet's theorem on ...
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
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.
Bertrand's box paradox – Mathematical paradox; Bertrand's postulate – Existence of a prime number between any number and its double; Bertrand's theorem – Physics theorem; Bertrand's ballot theorem – Theorem that gives the probability that an election winner will lead the loser throughout the count; Bertrand–Edgeworth model ...
Its value, using the modern definition of prime, [1] is approximately 0.235711131719232931374143... (sequence A033308 in the OEIS). The constant is irrational; this can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even integer is a sum of at most ...
An elementary proof of Bertrand's postulate on the existence of a prime in any interval of the form [,], one of the first results of Paul Erdős, was based on the divisibility properties of factorials.