Bertrand's postulate

Bertrand's postulate (actually a theorem) states that if "n" > 3 is an integer, then there always exists at least one prime number "p" with "n" < "p" < 2"n" − 2. A weaker but more elegant formulation is: for every "n" > 1 there is always at least one prime "p" such that "n" < "p" < 2"n".

This statement was first conjectured in 1845 by Joseph Bertrand (1822–1900). Bertrand himself verified his statement for all numbers in the interval [2, 3 &times; 106] .His conjecture was completely proved by Chebyshev (1821–1894) in 1850 and so the postulate is also called the Bertrand-Chebyshev theorem or Chebyshev's theorem. Ramanujan (1887&ndash;1920) used properties of the Gamma function to give a simpler proof [http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm] , from which the concept of Ramanujan primes would later arise, and Erdős (1913–1996) in 1932 published a simpler proof using the Chebyshev function vartheta(x), defined as:

: vartheta(x) = sum_{p=2}^{x} ln (p)

where "p" ≤ "x" runs over primes, and the binomial coefficients. See proof of Bertrand's postulate for the details.

Sylvester's theorem

Bertrand's postulate was proposed for applications to permutation groups. Sylvester (1814–1897) generalized it with the statement: the product of "k" consecutive integers greater than "k" is divisible by a prime greater than "k".

Erdős's theorems

Erdős proved that for any positive integer "k", there is a natural number "N" such that for all "n" > "N", there are at least "k" primes between "n" and 2"n". An equivalent statement had been proved earlier by Ramanujan (see Ramanujan prime).

The prime number theorem (PNT) suggests that the number of primes between "n" and 2"n" is roughly "n"/ln("n") when "n" is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's Postulate. That is, this theorem is comparatively weaker than the PNT. However, in order to use the PNT to prove results like Bertrand's Postulate, we would have to have very tight bounds on the error terms in the theorem—that is, we have to know fairly precisely what "roughly" means in the PNT. Such error estimates are available but are very difficult to prove (and the estimates are only sufficient for large values of "n"). By contrast, Bertrand's Postulate can be stated more memorably and proved more easily, and makes precise claims about what happens for small values of "n". (In addition, Chebyshev's theorem was proved before the PNT and so has historical interest.)

The similar and still unsolved Legendre's conjecture asks whether for every "n" > 1, there is a prime "p", such that "n"2 < "p" < ("n" + 1)2. Again we expect from the PNT that there will be not just one but many primes between "n"2 and ("n" + 1)2, but in this case the error estimates on the PNT are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval.

References

*
*MathWorld|urlname=BertrandsPostulate|title=Bertrand's Postulate|author=Jonathan Sondow and Eric W. Weisstein
*Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=BertrandsPostulate "Bertrand's postulate"] at Prime Pages glossary.
*
*


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Proof of Bertrand's postulate — In mathematics, Bertrand s postulate (actually a theorem) states that for each n ≥ 2 there is a prime p such that n < p < 2 n . It was first proven by Pafnuty Chebyshev, and a short but advanced proof was given by Srinivasa Ramanujan. The gist of …   Wikipedia

  • Joseph Louis François Bertrand — (March 11, 1822 – April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics.Bertrand was a professor at the École Polytechnique and… …   Wikipedia

  • Nombre premier de Ramanujan — En mathématiques, un nombre premier de Ramanujan est un nombre premier qui satisfait un résultat démontré par Srinivasa Ramanujan relatif à la fonction de compte des nombres premiers. Origines et définition En 1919, Ramanujan publia une nouvelle… …   Wikipédia en Français

  • List of number theory topics — This is a list of number theory topics, by Wikipedia page. See also List of recreational number theory topics Topics in cryptography Contents 1 Factors 2 Fractions 3 Modular arithmetic …   Wikipedia

  • Ramanujan prime — In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime counting function.Origins and definitionIn 1919, Ramanujan published a new proof [S. Ramanujan, A proof of Bertrand s… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • Prime number theorem — PNT redirects here. For other uses, see PNT (disambiguation). In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.