Largest square divisors of shifted primes
Runbo Li
TL;DR
The paper proves that there are infinitely many primes p for which p−a is divisible by a large square d^2 with d^2 ≥ p^{1/2+1/700} for any fixed nonzero a, refining prior work that reached p^{1/2+1/2000}. The authors employ Harman's sieve with a multilevel decomposition and a careful treatment of Type II information, using Buchstab's identity (and its reverse) to expose almost-primes and control error terms. The main achievement is a quantified reduction to lower bounds on sums over square-moduli and a final, explicit loss bound totaling less than 1, which yields the infinitude result. The work advances the square-divisor approach toward a resolution of Euler's conjecture by pushing the admissible exponent beyond 1/2 through refined sieve techniques and numerical bounds on integral contributions.
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.