On a conjectural congruence of Guo

TL;DR

This paper proves a conjecture by Guo that a specific product involving primes congruent to 3 mod 4 is congruent to 1 modulo p^2, advancing understanding of prime-related congruences.

Contribution

The paper confirms Guo's conjecture on a product congruence involving primes congruent to 3 mod 4, providing a new result in number theory.

Findings

The product is congruent to 1 modulo p^2 for primes p ≡ 3 mod 4.

The result holds for all positive integers r.

It advances the understanding of prime-related congruences.

Abstract

Let $p\equiv3\pmod{4}$ be a prime and $r$ a positive integer. We show that $$ \prod_{k=1}^{(p^{2r}-1)/2}\frac{4k-1}{4k+1}\equiv1\pmod{p^2}. $$ This confirms a recent conjecture of Guo.