A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes

TL;DR

This paper studies remainder problems for moduli that are products of distinct primes, focusing on sums of $L$-th powers. It proves that if $L$ is divisible by $\mathrm{lcm}(p_{1}-1,\dots,p_{n}-1)$ and $a_i$ satisfy $\gcd(a_i, p_1p_2\cdots p_n)=p_i$, then $a_{1}^{L}+\cdots+a_{n}^{L} \equiv n-1 \pmod{p_1p_2\cdots p_n}$; the special case $a_i=p_i$ yields the core congruence. The main contributions include Theorem T1 for $n=2$, Theorem T2 for $n=3$, and Theorem T3 which generalizes to arbitrary $n$, along with corollaries that connect to Euler-type results. The paper provides detailed proofs using Fermat’s and Euler’s theorems and CRT, plus concrete examples illustrating the results and their potential applicability to constructive remainder formulas in number theory and cryptography.

Abstract

Let p1, p2,..., pn be distinct prime numbers, and let Nn be their product. We prove that, for any positive integer L that is divisible by the least common multiple of p1 minus one, p2 minus one, and so on, and for integers a1, a2,..., an satisfying that each ai is relatively prime to Nn and shares the same prime factor pi, a certain congruence relation holds among their Lth powers.

Theorems & Definitions (16)

• Theorem 1: Fermat's little theorem
• Theorem 2: Euler's Theorem
• Example 1
• Example 2
• Definition 1
• Theorem 3
• Remark 1
• Corollary 1
• Corollary 2
• Theorem 4