# Bertrand's postulate

**Bertand's Postulate**states that if

*n*is an integer greater than 3, then there is at least one prime between

*n*and 2

*n*-2. It is often stated in the weaker, but perhaps more elegant form: If

*n*is a positive integer, then there is a prime

*p*with

*n*<

*p*≤ 2

*n*.

In 1845 Bertrand verified this for *n* less than
3,000,000, and in 1850 Tchebychef gave the first complete
proof. In 1932 Erdös gave a much nicer proof using the binomial
coefficients, which is the basis of the proof in most
modern texts (for example, theorem 418 of
[HW79]). This proof can be modified to prove that for
any positive integer *k*, there is a number *N*
such that for all *n* > *N*, there are *k*
primes between *n* and *2n*.

Notice Bertrand's postulate tells us that for any
prime *p*, there is another prime less than
2*p*, so we have the weak result that the
*n*th prime *p*_{n} is at most
2^{n}. This gives us the following way to
create a formula for the *n*th prime: Let *r*
be any integer greater than one, then let *a* be
the sum

Thep_{1}r^{-1}+p_{2}r^{-4}+p_{3}r^{-9}+ ... +p_{n}r^{ - n2}+ ...

*n*th prime is given by

This is cute, but computationally useless.p_{n}= floor(r^{n2}a) -r^{2n-1}floor(r^{ (n-1)2}a)

**See Also:** PrimeNumberThm, FormulasForPrimes

**References:**

- HW79 (chapter 22)
G. H. HardyandE. M. Wright,An introduction to the theory of numbers, Oxford University Press, 1979. ISBN 0198531702.MR 81i:10002(Annotation available)