# Bertrand's postulate

**Bertrand's postulate** is a theorem stating that for any integer , there always 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 numbers in the interval [2, 3 × 10^{6}].
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 , where is the prime counting function (number of primes less than or equal to ):

- , for all .

## Prime number theorem

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 2*x* then we see the number of primes up to 2*x* is asymptotically twice the number of primes up to *x* (the terms ln(2*x*) and ln(*x*) are asymptotically equivalent). Therefore 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. 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 *n*^{2} < *p* < (*n* + 1)^{2}. Again we expect that there will be not just one but many primes between *n*^{2} and (*n* + 1)^{2}, but in this case the PNT doesn't help: the number of primes up to *x*^{2} is asymptotic to *x*^{2}/ln(*x*^{2}) 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 *x*^{2}. So unlike the previous case of *x* and 2*x* 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.

## Generalizations

In 1919, Ramanujan (1887–1920) used properties of the Gamma function to give a simpler proof.[4] The 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 happened; for instance, there is a proof that

with *p*_{k} the *k*th prime and *R*_{n} the *n*th 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 2006, M. El Bachraoui proved that there exists a prime between 2*n* and 3*n*.[5] In 1973, Denis Hanson proved that there exists a prime between 3*n* and 4*n*.[6] Furthermore, in 2011, Andy Loo proved that as *n* tends to infinity, the number of primes between 3*n* and 4*n* also goes to infinity, thereby generalizing Erdős' and Ramanujan's results (see the section on Erdős' theorems below).[7] The first result is obtained with elementary methods. The second one is based on analytic bounds for the factorial function.

## Sylvester's theorem

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* = 2*n*, 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 2*n* is not prime, and thus indeed we now know there exists a prime *p* with *n* < *p* < 2*n*.

## Erdős's theorems

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

where *p* ≤ *x* runs over primes. See proof of Bertrand's postulate for the details.

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

## Better results

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

which implies that goes to infinity (and, in particular, is greater than 1 for sufficiently large ).[8]

Non-asymptotic bounds have also been proved. In 1952, Jitsuro Nagura proved that for there is always a prime between and .[9]

In 1976, Lowell Schoenfeld showed that for , there is always a prime in the open interval .[10]

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 .[11]

In 2010 Pierre Dusart proved that for there is at least one prime in the interval .[12]

In 2016, Pierre Dusart improved his result from 2010, showing (Proposition 5.4) that, if , there is at least one prime in the interval .[13] 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 .[14]

## Consequences

- The sequence of primes, along with 1, is a complete sequence; any positive integer can be written as a sum of primes (and 1) using each at most once.
- The only harmonic number that is an integer is the number 1.[15]

## See also

## Notes

- Ribenboim, Paulo (2004).
*The Little Book of Bigger Primes*. New York: Springer-Verlag. p. 181. ISBN 978-0-387-20169-6. - Joseph Bertrand. Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme. Journal de l'Ecole Royale Polytechnique, Cahier 30, Vol. 18 (1845), 123-140.
- P. Tchebychev. Mémoire sur les nombres premiers. Journal de mathématiques pures et appliquées, Sér. 1(1852), 366-390. (Proof of the postulate: 371-382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp.15-33, 1854
- Ramanujan, S. (1919). "A proof of Bertrand's postulate".
*Journal of the Indian Mathematical Society*.**11**: 181–182. - M. El Bachraoui, Primes in the Interval (2n, 3n)
- Hanson, Denis (1973), "On a theorem of Sylvester and Schur",
*Canadian Mathematical Bulletin*,**16**(2): 195–199. - Loo, Andy (2011), "On the Primes in the Interval (3
*n*, 4*n*)" (PDF),*International Journal of Contemporary Mathematical Sciences*,**6**(38): 1871–1882 - G. H. Hardy and E. M. Wright,
*An Introduction to the Theory of Numbers*, 6th ed., Oxford University Press, 2008, p. 494. - 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. - 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. - Dusart, Pierre (1998),
*Autour de la fonction qui compte le nombre de nombres premiers*(PDF) (PhD thesis) (in French) - Dusart, Pierre (2010). "Estimates of Some Functions Over Primes without R.H.". arXiv:1002.0442 [math.NT].
- Dusart, Pierre (2016). "Explicit estimates of some functions over primes".
*The Ramanujan Journal*.**45**: 227–251. doi:10.1007/s11139-016-9839-4. - 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 10.1.1.360.3671. doi:10.1112/plms/83.3.532. - Ronald L., Graham; Donald E., Knuth; Oren, Patashnik (1994).
*Concrete Mathematics*. Addison-Wesley.

## Bibliography

- P. Erdős (1934). "A Theorem of Sylvester and Schur".
*Journal of the London Mathematical Society*.**9**(4): 282–288. doi:10.1112/jlms/s1-9.4.282. - Jitsuro Nagura (1952). "On the interval containing at least one prime number".
*Proc. Japan Acad*.**28**(4): 177–181. doi:10.3792/pja/1195570997. - Chris Caldwell,
*Bertrand's postulate*at Prime Pages glossary. - H. Ricardo (2005). "Goldbach's Conjecture Implies Bertrand's Postulate".
*Amer. Math. Monthly*.**112**: 492. - Hugh L. Montgomery; Robert C. Vaughan (2007).
*Multiplicative number theory I. Classical theory*. Cambridge tracts in advanced mathematics.**97**. Cambridge: Cambridge Univ. Press. p. 49. ISBN 978-0-521-84903-6. - J. Sondow (2009). "Ramanujan primes and Bertrand's postulate".
*Amer. Math. Monthly*.**116**(7): 630–635. arXiv:0907.5232. doi:10.4169/193009709x458609.

## External links

- Sondow, Jonathan and Weisstein, Eric W. "Bertrand's Postulate".
*MathWorld*.CS1 maint: multiple names: authors list (link) - A proof of the weak version in the Mizar system: http://mizar.org/version/current/html/nat_4.html#T56