Largest square divisors of shifted primes

TL;DR

This paper proves infinitely many primes p where p-a is divisible by large squares, improving previous bounds using advanced sieve techniques, thus advancing understanding of prime divisibility properties.

Contribution

It improves the known lower bound on the size of squares dividing p-a for infinitely many primes p, using Harman's sieve methods.

Findings

Infinitely many primes p with large square divisors of p-a

Improved the exponent bound from 1/2 + 1/2000 to 1/2 + 1/700

Applied advanced sieve techniques to prime divisibility problems

Abstract

The author shows that there are infinitely many primes $p$ such that for any nonzero integer $a$, $p-a$ is divisible by a square $d^2 > p^{\frac{1}{2}+\frac{1}{700}}$. The exponent $\frac{1}{2}+\frac{1}{700}$ improves Merikoski's $\frac{1}{2}+\frac{1}{2000}$. Many powerful devices in Harman's sieve are used for this improvement.