Bertrand's postulate

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Joseph Louis François Bertrand

In number theory, Bertrand's postulate is the theorem that for any integer , there exists at least one prime number with

A less restrictive formulation is: for every , there is always at least one prime such that

Another formulation, where is the -th prime, is: for


This statement was first conjectured in 1845 by Joseph Bertrand[2] (1822–1900). Bertrand himself verified his statement for all integers .

His conjecture was completely proved by Chebyshev (1821–1894) in 1852[3] 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 , the prime-counting function (number of primes less than or equal to ):

, for all .

Prime number theorem[edit]

The prime number theorem (PNT) implies that the number of primes up to x is roughly x/ln(x), so if we replace x with 2x then we see the number of primes up to 2x is asymptotically twice the number of primes up to x (the terms ln(2x) and ln(x) are asymptotically equivalent). Therefore, the number of primes between n and 2n 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. So Bertrand's postulate is comparatively weaker than the PNT. But PNT is a deep theorem, while Bertrand's Postulate can be stated more memorably and proved more easily, and also 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 n2 < p < (n + 1)2. Again we expect that there will be not just one but many primes between n2 and (n + 1)2, but in this case the PNT doesn't help: the number of primes up to x2 is asymptotic to x2/ln(x2) while the number of primes up to (x + 1)2 is asymptotic to (x + 1)2/ln((x + 1)2), which is asymptotic to the estimate on primes up to x2. So unlike the previous case of x and 2x we don't get a proof of Legendre's conjecture even for all large n. Error estimates on the PNT are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval.


In 1919, Ramanujan (1887–1920) used properties of the Gamma function to give a simpler proof than Chebyshev's.[4] His short paper included a generalization of the postulate, from which would later arise the concept of Ramanujan primes. Further generalizations of Ramanujan primes have also been discovered; for instance, there is a proof that

with pk the kth prime and Rn the nth Ramanujan prime.

Other generalizations of Bertrand's postulate have been obtained using elementary methods. (In the following, n runs through the set of positive integers.) In 1973, Denis Hanson proved that there exists a prime between 3n and 4n.[5] In 2006, apparently unaware of Hanson's result, M. El Bachraoui proposed a proof that there exists a prime between 2n and 3n.[6]

Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane. Thus, as Gaussian primes extend over the plane and not only along a line, and doubling a complex number is not simply multiplying by 2 but doubling its norm (multiplying by 1+i), different definitions lead to different results, some are still conjectures, some proven.[7]

Sylvester's theorem[edit]

Bertrand's postulate was proposed for applications to permutation groups. Sylvester (1814–1897) generalized the weaker statement with the statement: the product of k consecutive integers greater than k is divisible by a prime greater than k. Bertrand's (weaker) postulate follows from this by taking k = n, and considering the k numbers n + 1, n + 2, up to and including n + k = 2n, where n > 1. According to Sylvester's generalization, one of these numbers has a prime factor greater than k. Since all these numbers are less than 2(k + 1), the number with a prime factor greater than k has only one prime factor, and thus is a prime. Note that 2n is not prime, and thus indeed we now know there exists a prime p with n < p < 2n.

Erdős's theorems[edit]

In 1932, Erdős (1913–1996) also published a simpler proof using binomial coefficients and the Chebyshev function , defined as:

where px runs over primes. See proof of Bertrand's postulate for the details.[8]

Erdős proved in 1934 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 2n. An equivalent statement had been proved in 1919 by Ramanujan (see Ramanujan prime).

Better results[edit]

It follows from the prime number theorem that for any real there is a such that for all there is a prime such that . It can be shown, for instance, that

which implies that goes to infinity (and, in particular, is greater than 1 for sufficiently large ).[9]

Non-asymptotic bounds have also been proved. In 1952, Jitsuro Nagura proved that for there is always a prime between and .[10]

In 1976, Lowell Schoenfeld showed that for , there is always a prime in the open interval .[11]

In his 1998 doctoral thesis, Pierre Dusart improved the above result, showing that for , , and in particular for , there exists a prime in the interval .[12]

In 2010 Pierre Dusart proved that for there is at least one prime in the interval .[13]

In 2016, Pierre Dusart improved his result from 2010, showing (Proposition 5.4) that if , there is at least one prime in the interval .[14] He also shows (Corollary 5.5) that for , there is at least one prime in the interval .

Baker, Harman and Pintz proved that there is a prime in the interval for all sufficiently large .[15]

Dudek proved that for all , there is at least one prime between and .[16]

Dudek also proved that the Riemann hypothesis implies that for all there is a prime satisfying



See also[edit]


  1. ^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 181. ISBN 978-0-387-20169-6.
  2. ^ Bertrand, Joseph (1845), "Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme.", Journal de l'École Royale Polytechnique (in French), 18 (Cahier 30): 123–140.
  3. ^ Tchebychev, P. (1852), "Mémoire sur les nombres premiers." (PDF), Journal de mathématiques pures et appliquées, Série 1 (in French): 366–390. (Proof of the postulate: 371-382). Also see Tchebychev, P. (1854), "Mémoire sur les nombres premiers.", Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg (in French), 7: 15–33
  4. ^ Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society, 11: 181–182
  5. ^ Hanson, Denis (1973), "On a theorem of Sylvester and Schur", Canadian Mathematical Bulletin, 16 (2): 195–199, doi:10.4153/CMB-1973-035-3.
  6. ^ El Bachraoui, Mohamed (2006), "Primes in the interval [2n,3n]", International Journal of Contemporary Mathematical Sciences, 1
  7. ^ Madhuparna Das (2019), Generalization of Bertrand’s postulate for Gaussian primes, arXiv:1901.07086v2
  8. ^ Erdős, P. (1932), "Beweis eines Satzes von Tschebyschef" (PDF), Acta Litt. Sci. (Szeged) (in German), 5 (1930-1932): 194–198
  9. ^ G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
  10. ^ Nagura, J (1952), "On the interval containing at least one prime number", Proceedings of the Japan Academy, Series A, 28 (4): 177–181, doi:10.3792/pja/1195570997
  11. ^ Lowell Schoenfeld (April 1976), "Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x), II", Mathematics of Computation, 30 (134): 337–360, doi:10.2307/2005976, JSTOR 2005976
  12. ^ Dusart, Pierre (1998), Autour de la fonction qui compte le nombre de nombres premiers (PDF) (PhD thesis) (in French)
  13. ^ Dusart, Pierre (2010). "Estimates of Some Functions Over Primes without R.H.". arXiv:1002.0442 [math.NT].
  14. ^ Dusart, Pierre (2016), "Explicit estimates of some functions over primes", The Ramanujan Journal, 45: 227–251, doi:10.1007/s11139-016-9839-4, S2CID 125120533
  15. ^ Baker, R. C.; Harman, G.; Pintz, J. (2001), "The difference between consecutive primes, II", Proceedings of the London Mathematical Society, 83 (3): 532–562, CiteSeerX, doi:10.1112/plms/83.3.532, S2CID 8964027
  16. ^ Dudek, Adrian (December 2016), "An explicit result for primes between cubes", Funct. Approx., 55 (2): 177–197, arXiv:1401.4233, doi:10.7169/facm/2016.55.2.3, S2CID 119143089
  17. ^ Dudek, Adrian W. (21 August 2014), "On the Riemann hypothesis and the difference between primes", International Journal of Number Theory, 11 (3): 771–778, arXiv:1402.6417, Bibcode:2014arXiv1402.6417D, doi:10.1142/S1793042115500426, ISSN 1793-0421, S2CID 119321107
  18. ^ Ronald L., Graham; Donald E., Knuth; Oren, Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley. ISBN 978-0-201-55802-9.


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