Can anyone sketch the proof or provide a link that there is always a prime between $n^3$ and $(n+1)^3$

Question

In a recent forum discussion on number theory, it was mentioned that A. E. Ingham had proven that there is always a prime between $n^3$ and $(n+1)^3$.

Does anyone know if there is a link available on the web or knows a rough sketch of the proof. Does it use sieve theory?

I am very interested in checking out the proof.

Answer 1

I'd venture the discussion in the fora was mistaken.

As late as 2014, it appears the best is a bound on where this is true, from

An Explicit Result for Primes Between Cubes - A. Dudek

Answer 2

Back in the mid-80s, when I first opened Apostol's "Introduction to Analytic Number Theory," Apostol stated the theorem that there exists a real $\alpha$ such that $\left\lfloor\alpha^{3^n}\right\rfloor$ was always prime. Apostol noted, though, that the existing proof was non-constructive.

I realized relatively quickly that if you could prove there was always a prime between $n^3$ and $(n+1)^3$, you'd have an easy constructive proof.

So I highly doubt that there was a proof before 1967, when Wikipedia says Ingham died. Apostol's book was first published in 1976.