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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.
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
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 ...
Erdős found a proof for Bertrand's postulate which proved to be far neater than Chebyshev's original one. He also discovered the first elementary proof for the prime number theorem, along with Atle Selberg. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg.
Copeland and Erdős's proof that their constant is normal relies only on the fact that is strictly increasing and = + (), where is the n th prime number. More generally, if is any strictly increasing sequence of natural numbers such that = + and is any natural number greater than or equal to 2, then the constant obtained by concatenating "0."
The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3.; The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set.
The binary conjecture is sufficient for Bertrand's postulate, whereas Bertrand's postulate is necessary for the binary conjecture. Also, the binary conjecture and the ternary conjecture are equivalent. If one is true, so is the other. 2605:E000:6116:7D00:4CD6:5569:EA6F:731C 14:40, 4 October 2017 (UTC)
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.