About: Proof of Bertrand's postulate

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In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The following elementary proof was published by Paul Erdős in 1932, as one of his earliest mathematical publications. The basic idea is to show that the central binomial coefficients need to have a prime factor within the interval in order to be large enough. This is achieved through analysis of the factorization of the central binomial coefficients.

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rdfs:label	Dimostrazione del postulato di Bertrand (it) Proof of Bertrand's postulate (en)
rdfs:comment	In matematica, il postulato di Bertrand afferma che per ogni n ≥ 2 esiste un primo p tale che n < p < 2n. La prima dimostrazione fu data da Pafnuty Chebyshev; (it) In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The following elementary proof was published by Paul Erdős in 1932, as one of his earliest mathematical publications. The basic idea is to show that the central binomial coefficients need to have a prime factor within the interval in order to be large enough. This is achieved through analysis of the factorization of the central binomial coefficients. (en)
dcterms:subject	Theorems about prime numbers Factorial and binomial topics Article proofs Prime numbers
Wikipage page ID	254237 (xsd:integer)
Wikipage revision ID	1117107844 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime power Primorial Elementary proof Mizar system Binomial theorem Bertrand's postulate Theorems about prime numbers Paul Erdős Factorial and binomial topics Article proofs Mathematical induction Mathematics Prime factor Concave function Theorem Logarithm Parity (mathematics) Prime numbers P-adic order Central binomial coefficient Legendre's formula Lemma (mathematics) Proof by contradiction Mathematical proof Interval (mathematics) Counterexample Prime number Asymptotic analysis Chebyshev Integer Natural number Real number Ramanujan
Link from a Wikipage to an external page	http://mizar.org/version/current/html/nat_4.html%23T56 https://web.williams.edu/Mathematics/lg5/Chebyshev.pdf https://sites.math.washington.edu/~mathcircle/circle/2013-14/advanced/mc-13a-w10.pdf
sameAs	Proof of Bertrand's postulate Proof of Bertrand's postulate Proof of Bertrand's postulate Proof of Bertrand's postulate Proof of Bertrand's postulate
dbp:wikiPageUsesTemplate	dbt:Mathworld dbt:Reflist
title	Bertrand's Postulate (en)
urlname	BertrandsPostulate (en)
has abstract	In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The following elementary proof was published by Paul Erdős in 1932, as one of his earliest mathematical publications. The basic idea is to show that the central binomial coefficients need to have a prime factor within the interval in order to be large enough. This is achieved through analysis of the factorization of the central binomial coefficients. The main steps of the proof are as follows. First, show that the contribution of every prime power factor in the prime decomposition of the central binomial coefficient is at most . Then show that every prime larger than appears at most once. The next step is to prove that has no prime factors in the interval . As a consequence of these bounds, the contribution to the size of coming from the prime factors that are at most grows asymptotically as for some . Since the asymptotic growth of the central binomial coefficient is at least , the conclusion is that, by contradiction and for large enough , the binomial coefficient must have another prime factor, which can only lie between and . The argument given is valid for all . The remaining values of are by direct inspection, which completes the proof. (en) In matematica, il postulato di Bertrand afferma che per ogni n ≥ 2 esiste un primo p tale che n < p < 2n. La prima dimostrazione fu data da Pafnuty Chebyshev; (it)
prov:wasDerivedFrom	wikipedia-en:Proof_of_Bertrand's_postulate?oldid=1117107844&ns=0
page length (characters) of wiki page	10301 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Proof_of_Bertrand's_postulate
is Link from a Wikipage to another Wikipage of	Bertrand's postulate List of mathematical proofs List of number theory topics Euclid's theorem Central binomial coefficient Proofs from THE BOOK List of factorial and binomial topics Factorial
is foaf:primaryTopic of	wikipedia-en:Proof_of_Bertrand's_postulate

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