Bertrand's postulate
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Bertrand's postulate states that if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2n − 2.
This statement was first made in 1845 by Joseph Bertrand.[1] Bertrand verified his statement for all numbers in the interval [2, 3 × 106].
His statement was completely proven by Pafnuty Chebyshev in 1850.[2] For this reason, the postulate is also called the Bertrand-Chebyshev theorem or Chebyshev's theorem. Srinivasa Ramanujan gave a simpler proof. Ramanujan later used that proof when he discovered Ramanujan primes. In 1932, Paul Erdős published a simpler proof using the Chebyshev function θ(x).[3]
References
[change | change source]- ↑ Journal de l'ecole imperiale politechnique (in French). 1845.
- ↑ Chebyshev, Pafnuty. "Mémoire sur les nombres premiers" (PDF). Journal de Mathématiques Pures et Appliquées.
- ↑ Erdos, Paul (1932). "Beweis eines Satzes von Tschebyschef" (PDF). Acta Scientarium Mathematicarum.
