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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.

Larry Freeman
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  • Did they say when it had been proven? – Thomas Andrews Jan 17 '15 at 02:12
  • It must have been a while ago. Wikipedia reports that he died in 1967. – Larry Freeman Jan 17 '15 at 02:13
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    In Apostol's analytic number theory book, he states the theorem that there is a real number $\alpha$ such that $\lfloor\alpha^{3^n}\rfloor$ is always a prime, but that the proof is non-constructive. But this theorem would give an easy constructive method to find $\alpha$ - actually, an arbitrary number of $\alpha$s. – Thomas Andrews Jan 17 '15 at 02:14

2 Answers2

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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

HammyTheGreek
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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.

Thomas Andrews
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  • Thanks very much. I did a Google search before posting my question and couldn't find anything. – Larry Freeman Jan 17 '15 at 02:32
  • It's certainly not a conclusive answer, but it seems unlikely. When I first read your question, I thought it might be referencing a new result, but given the death in 1967, that seems unlikely. – Thomas Andrews Jan 17 '15 at 02:39
  • Ingham's proof is mentioned in HammyTheGreek's link. See [7]. – Larry Freeman Jan 17 '15 at 02:55
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    It might well be that Ingham's result is the result Apostol was referring to - that Ingham knew there were primes between consecutive cubes for sufficiently large cubes, but had no bound for "sufficiently large." @LarryFreeman – Thomas Andrews Jan 17 '15 at 03:00
  • That's exactly it. Ingham proved that there were primes for sufficiently large cubes and had no bound. :-) – Larry Freeman Jan 17 '15 at 03:03