Editing Bertrand's postulate (section)

[[File:Bertrand.jpg|thumb|[[Joseph Louis François Bertrand]]]]
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{{Short description|Existence of a prime number between any number and its double}}
In [[number theory]], '''Bertrand's postulate''' is the [[theorem]] that for any [[integer]] <math>n > 3</math>, there exists at least one [[prime number]] <math>p</math> with 
:<math>n < p < 2n - 2.</math>
A less restrictive formulation is: for every <math>n > 1</math>, there is always at least one prime <math>p</math> such that 
:<math>n < p < 2n.</math>
Another formulation, where <math>p_n</math> is the <math>n</math>-th prime, is: for <math>n \ge 1</math>
:<math> p_{n+1} < 2p_n.</math><ref>{{cite book|last=Ribenboim|first=Paulo|title=The Little Book of Bigger Primes|url=https://archive.org/details/littlebookbigger00ribe_610|url-access=limited|date=2004|publisher=Springer-Verlag|location=New York|isbn=978-0-387-20169-6|page=[https://archive.org/details/littlebookbigger00ribe_610/page/n205 181]}}</ref>

This statement was first [[conjecture]]d in 1845 by [[Joseph Bertrand]]<ref>{{Citation|first=Joseph |last=Bertrand |author-link=Joseph Bertrand |title=Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme. |journal=Journal de l'École Royale Polytechnique |issue=Cahier 30 |volume=18 |year=1845 |pages=123–140 |language=fr |url={{Google Books|WTa-qRIWckoC|page=123|plainurl=yes}}}}.</ref> (1822–1900). Bertrand himself verified his statement for all integers <math>2 \le n \le 3\,000\,000</math>.

His conjecture was completely [[Proof of Bertrand's postulate|proved]] by [[Pafnuty Chebyshev|Chebyshev]] (1821–1894) in 1852<ref>{{Citation|first=P. |last=Tchebychev |author-link=Pafnuty Chebyshev |title=Mémoire sur les nombres premiers. |journal=Journal de mathématiques pures et appliquées |series=Série 1 |year=1852 |pages=366–390 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf |language=fr}}. (Proof of the postulate: 371-382). Also see {{Citation|first=P. |last=Tchebychev |author-link=Pafnuty Chebyshev |title=Mémoire sur les nombres premiers. |journal=Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg |volume=7 |year=1854 |pages=15-33 |url=https://www.biodiversitylibrary.org/page/37114947 |language=fr}}</ref> and so the postulate is also called the '''Bertrand–Chebyshev theorem''' or '''Chebyshev's theorem'''.  Chebyshev's theorem can also be stated as a relationship with <math>\pi(x)</math>, the [[prime-counting function]] (number of primes less than or equal to <math>x</math>):
:<math>\pi(x) - \pi\bigl(\tfrac{x}{2}\bigr) \ge 1, \text{ for all } x \ge 2.</math>