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<caldwell@utm.edu> Bertand's Postulate states that if n is an integer greater than 3, then there is at least one prime between n and 2n2. 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 < 2n. 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 2p, so we have the weak result that the nth prime p_{n} is at most 2^{n}. This gives us the following way to create a formula for the nth prime: Let r be any integer greater than one, then let a be the sum p_{1}r^{1} + p_{2}r^{4} + p_{3}r^{9} + ... + p_{n}r^{  n2} + ...The nth prime is given by p_{n} = floor(r^{n2}a)  r^{2n1}floor(r^{ (n1)2}a)This is cute, but computationally useless.
See Also: PrimeNumberThm, FormulasForPrimes Related pages (outside of this work) References:
Chris K. Caldwell © 19992014 (all rights reserved)
