Bertrand's postulate

For every integer n greater than 1, there is a prime number p satisfying n < p < 2n. It is also called Bertrand's theorem.

Binomial coefficient: A coefficient counting ways to choose a fixed-size subset, appearing in the expansion of a power of a sum. Central binomial coefficients encode prime factors that force a prime between n and 2n.
Joseph Bertrand: A French mathematician known for work in probability, algebra, and number theory. He stated the prime-existence claim in 1845 after checking numerical cases.
Prime gap: The difference between consecutive prime numbers. Bertrand's bound limits how large a prime-free interval can be in a particular range.
Legendre's conjecture: The conjecture that every interval between consecutive squares contains a prime. It proposes prime existence in shorter, differently positioned intervals and remains unproved.
Central binomial coefficient: The middle coefficient in a row of Pascal's triangle, written as binomial(2n,n). Its size provides the lower bound that drives Erdős's proof.
Pafnuty Chebyshev: A Russian mathematician whose work established foundational estimates for the distribution of primes. He proved Bertrand's claim in 1852 using analytic estimates on primes.
Bertrand's postulate in algorithm analysis: Use of Bertrand's postulate to guarantee a prime within a factor of two of an integer parameter. Such guarantees can establish the existence of prime-sized parameters without locating them.
Andrica's conjecture: The conjecture that the square roots of consecutive primes differ by less than one. It is another unproved constraint on prime gaps, rather than a proved factor-two bound.
Prime factorization: The expression of an integer as a product of primes, unique up to the order of factors. Bounding prime powers in binomial coefficients exposes primes within the required interval.
Paul Erdős: A Hungarian mathematician known for prolific collaborations and contributions across discrete mathematics and number theory. His 1932 proof made the postulate's proof notably elementary.

Linked from 10 pages