Some curious congruences modulo primes

TL;DR

This paper investigates specific congruences involving sums over prime moduli, revealing new properties of these sums related to prime numbers and modular arithmetic.

Contribution

It derives a novel congruence involving sums over multiple indices modulo primes, expanding understanding of prime-related congruences.

Findings

Established a new congruence for sums over indices modulo primes

Demonstrated the sum equals zero modulo p under given conditions

Provides insights into the structure of sums involving primes and modular arithmetic

Abstract

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence: $$\sum_{0<i_1<...<i_n<p}(i_1/3)(-1)^{i_1}/(i_1...i_n)=0 (mod p).$$