Bertrand's postulate

The Bertrand's postulate (also set by Bertrand-Chebyshev ) is a mathematical theorem , which states that for every natural number at least one prime number with exists. ${\displaystyle n>1}$ ${\displaystyle p}$ ${\displaystyle n<p<2\,n}$

This claim was first made in 1845 by the mathematician Joseph Bertrand , who proved it for natural numbers up to 3,000,000. Chebyshev provided the first complete proof of all natural numbers five years later. Another, simpler proof was given by the Indian mathematician S. Ramanujan , who also introduced Ramanujan prime numbers . In 1932, Paul Erdős also provided simple evidence.

Ramanujan proved a generalization, the existence of Ramanujan prime numbers , so that for all between and at least prime numbers lie. ${\displaystyle R_{n}}$ ${\displaystyle x\geq R_{n}}$ ${\displaystyle x}$ ${\displaystyle {\frac {x}{2}}}$ ${\displaystyle n}$

Proof for n ≤ 4000

For the first 4000 natural numbers, prime numbers can simply be given, so that the claim holds. As a result

${\displaystyle 2,3,5,7,13,23,43,83,163,317,631,1259,2503,4001}$(Follow A295262 in OEIS )

of prime numbers, each term in the sequence is less than twice the previous one. Thus the claim holds for ${\displaystyle n\leq 4000.}$

literature

• The BOOK of Evidence . Springer, Berlin 2002, ISBN 3-540-42535-7 (3rd edition: ISBN 978-3-642-02258-6 ).

Individual evidence

1. ↑ J. Bertrand : Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme. In: Journal de l'École Royale Polytechnique. 30 (18), 1845, pp. 123-140 (French).
2. ↑ Chef de cuisine : Mémoire sur les nombres premiers. (1850), Mémoires de l'académie impériale des sciences de St.-Pétersbourg 7, 1854, pp. 17-33; Journal de mathématiques pures et appliquées 1 ^re série 17, 1852, pp. 366-390.
   In: A. Markoff , N. Sonin (eds.): Oeuvres de P. L. Tchebychef. Tome I. St.-Pétersbourg 1899, pp. 51–70 (French; in the Internet archive ).
3. ^ S. Ramanujan : A proof of Bertrand's postulate. In: Journal of the Indian Mathematical Society. 11, 1919, pp. 181-182 (English).

Web links

• Eric W. Weisstein : Bertrand's Postulates . In: MathWorld (English).