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

TL;DR

The paper establishes effective bounds for primes in arithmetic progressions within short intervals and applies these results to prove that every large integer beyond a certain size can be expressed as a sum of seven cubes.

Contribution

It provides explicit bounds for the least prime in arithmetic progressions within short intervals and applies these bounds to the seven cubes problem.

Findings

Bound for the least prime in arithmetic progressions: P(a,q) ≤ e^{α (log q)^2}

Interval [e^x, e^{x+ε}] contains primes in arithmetic progressions for large x

Every integer larger than e^{71,000} can be written as a sum of seven cubes

Abstract

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$ (computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[ e^x,e^{x+\epsilon }\right]$ contains a prime $p$ in the arithmetic progression $a \bmod q$. This gives the bound for the least prime in this arithmetic progression: $P(a,q) \le e^{\alpha (\log q)^2}$. For instance for all $q\ge 10^{30}$, $P(a,q) \le e^{4.401(\log q)^2}$. Finally, we apply this result to establish that every integer larger than $e^{71\,000}$ is a sum of seven cubes.