Bertrand's postulate

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Bertrand's postulate (actually a theorem) states that for any integer n > 3, there always exists at least one prime number p with

n < p < 2n - 2.

A weaker but more elegant formulation is: for every n > 1 there is always at least one prime p such that

n < p < 2n.

Another formulation, where p_n is the n-th prime, is for n \ge 1

p_{n+1} < 2p_n.[1]

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 × 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. Chebyshev's theorem can also be stated as a relationship with \scriptstyle \pi(x) \,, where \scriptstyle \pi(x) \, is the prime counting function (number of primes less than or equal to \scriptstyle x \,):

\pi(x) - \pi(\tfrac{x}{2}) \ge 1,\, for all \, x \ge 2. \,

In 1919, Ramanujan (1887–1920) used properties of the Gamma function to give a simpler proof,[2] from which the concept of Ramanujan primes would later arise, and Erdős (1913–1996) in 1932 published a simpler proof using binomial coefficients and the Chebyshev function ϑ, defined as:

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

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

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.

Erdős's theorems[edit]

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).

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.

Better results[edit]

It follows from the prime number theorem that for any real \epsilon > 0 \text{, there exists an } n > n_0 such that there is always a prime between n and (1 + \epsilon) n \text{ for all } n > n_0: it can be shown, for instance, that

\lim_{n \to \infty}\frac{\pi((1+\epsilon)n)-\pi(n)}{n/\log n}=\epsilon,

which implies that  \pi (( 1 + \epsilon ) n) - \pi (n) goes to infinity (and, in particular, is greater than 1 for sufficiently large n).[3]

Non-asymptotic bounds have also been proved. In 1952, Jitsuro Nagura proved that for n ≥ 25, there is always a prime between n and (1 + 1/5)n.[4]

In 1976, Lowell Schoenfeld showed that for n ≥ 2010760, there is always a prime between n and (1 + 1/16597)n.[5]

In 1998, Pierre Dusart improved the result in his doctoral thesis, showing that for k ≥ 463, pk+1 ≤ (1 + 1/(ln2pk))pk, and in particular for x ≥ 3275, there exists a prime number between x and (1 + 1/(2ln2x))x.[6] In 2010 he proved, that for x ≥ 396738 there is at least one prime between x and (1 + 1/(25ln2x))x.[7]

Baker, Harman and Pintz proved that there is a prime in the interval [x,\,x+O(x^{21/40})] for all large x.[8]

Generalizations of Bertrand's Postulate have also been obtained by 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 2n and 3n.[9] In 2011, Andy Loo proved that there exists a prime between 3n and 4n. Furthermore, he proved that as n tends to infinity, the number of primes between 3n and 4n also goes to infinity, thereby generalizing Erdős' and Ramanujan's results (see the section on Erdős' theorems above).[10] None of these proofs require the use of deep analytic results.

In 2009, Nicholson proved that

2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,

i.e. pk is the kth prime and the nth Ramanujan prime. This means that there is at least one prime between p_k and 2p_{i-n}. The prime p_{i-n} is the next prime after R_n/2 and p_i is the smallest prime of this type for each k.

Consequences[edit]

  • 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 number 1 is the only integer which is a harmonic number.

See also[edit]

Notes[edit]

  1. ^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 181. ISBN 0-387-20169-6. 
  2. ^ Ramanujan, S. (1919). "A proof of Bertrand's postulate". Journal of the Indian Mathematical Society 11: 181–182. 
  3. ^ G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
  4. ^ Nagura, J. "On the interval containing at least one prime number." Proceedings of the Japan Academy, Series A 28 (1952), pp. 177–181.[1]
  5. ^ 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. 
  6. ^ Dusart, Pierre (1998), Autour de la fonction qui compte le nombre de nombres premiers (PDF) (in French) 
  7. ^ Dusart, Pierre (2010). "Estimates of Some Functions Over Primes without R.H.". arXiv:1002.0442.
  8. ^ Baker, R. C.; Harman, G.; Pintz, G.; Pintz, J. (2001). "The difference between consecutive primes, II". Proceedings of the London Mathematical Society 83 (3): 532–562. doi:10.1112/plms/83.3.532. 
  9. ^ M. El Bachraoui, Primes in the Interval (2n, 3n)
  10. ^ Loo, Andy (2011), On the Primes in the Interval (3n, 4n) (PDF), International Journal of Contemporary Mathematical Sciences 6 (38): 1871–1882 

Bibliography[edit]