Primes between consecutive cubes 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?
 A: 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 $n\ge\exp(\exp(33.217))$.
See also A060199 in the OEIS.
[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
A: This paper written fairly recently, and some of its sources might be useful: http://arxiv.org/pdf/0810.2113v2.pdf
A: We know since Ingham that
$$
p_{n+1} - p_n < K p_n^{5/8},
$$
where $p_n$ is the $n^{\rm th}$ prime and $K$ is a positive integer constant. Now let $N>K^8$ be a positive integer and $p_n$ the greatest prime smaller than $N^3$. Then we have:
$$
\begin{align}
N^3 &< p_{n+1} \\
&< K p_n^{5/8} + p_n \\
&< N^3 + KN^{15/8} \\
&< N^3 + N^2 \ \ \ \text{(since $N>K^8$)} \\
&< N^3 +3N^2 +3N 
\end{align}
$$
And now using the last line of the above reasoning we find that
$$
N^3 < p_{n+1} < \left({N + 1}\right)^3 - 1.
$$
