# On the largest square divisor of shifted primes

## TL;DR

This paper proves the existence of infinitely many primes p where p-1 has a large square divisor, improving previous bounds by applying advanced sieve methods and equidistribution estimates for smooth square moduli.

## Contribution

It introduces a new bilinear equidistribution estimate modulo smooth square moduli, surpassing the 1/2 barrier in the size of the square divisor of shifted primes.

## Key findings

- Infinitely many primes p with p-1 divisible by a large square d^2 ≥ p^θ.
- Enhanced bounds on the size of square divisors of p-1 for primes p.
- Application of Harman's sieve and advanced exponential sum estimates.

## Abstract

We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.$ This improves the work of Matom\"aki (2009) who obtained the result for $\theta=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $\theta=4/9-\varepsilon.$ Similarly as in the work of Matom\"aki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $\theta=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as Zhang (2014) used to obtain equidistribution beyond the Bombieri-Vinogradov range for primes with respect to smooth moduli. To optimize the argument we incorporate technical refinements from the Polymath project (2014). Since the moduli are squares, the method produces complete exponential sums modulo squares of primes which are estimated using the results of Cochrane and Zheng (2000).