Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

Kadiri, H.

doi:10.1090/S0025-5718-08-02084-X

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem
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Math. Comp. 77 (2008), 1733-1748 Request permission

Abstract:

For any $\epsilon >0$ and any non-exceptional modulus $q\ge 3$, we prove that, for $x$ large enough ($x\ge \alpha _{\epsilon }\log ^2 q$), the interval $\left [ e^x,e^{x+\epsilon }\right ]$ contains a prime $p$ in any of the arithmetic progressions modulo $q$. We apply this result to establish that every integer $n$ larger than $\exp (71 000)$ is a sum of seven cubes.

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