I am looking at Dudley's proof of the existence of Mill's constant. It starts out as follows
The proof depends on the following theorem: there is an integer $A$ such that if $n>A$, then there is a prime $p$ such that $$ n^3 < p < (n+1)^3 -1.$$ We will not prove this but we will use it to determine a sequence of primes$\ldots$
Where can I find a proof of this theorem?
On
Ingham [1] proved that for all large enough $n$ there is a prime $p$ such that $n^3<p<(n+1)^3.$ That this holds for all $n>0$ is not currently known. Cheng claims a proof, but Dudek disputes this:
We should note that a result has been given by Cheng [2], in which he purports to show the above theorem for the range $n\ge\exp(\exp(15))$. We should, however, note that he incorrectly goes from $$ n^3\ge\exp(\exp(45)) $$ to $$ n\ge\exp(\exp(15)) $$ in establishing his result. There are some other errors also, notably in his proof of Theorem 3 in his paper [2], the first inequality sign is backwards and he has used Chebyshev's $\psi$-function instead of the $\theta$-function.
The best result I know is in the same paper, proving that there is a prime between $n^3$ and $(n+1)^3$ for $n\ge\exp(\exp(33.217))$.
[1] A. E. Ingham, On the difference between consecutive primes, Quarterly Journal of Mathematics 1 (1937), pp. 255-266.
[2] Adrian Dudek, An Explicit Result for Primes Between Cubes
This paper written fairly recently, and some of its sources might be useful: http://arxiv.org/pdf/0810.2113v2.pdf