Sums of products of congruence classes and of arithmetic progressions

TL;DR

This paper characterizes the sum of products of congruence classes and arithmetic progressions, showing they form specific classes or progressions under certain gcd conditions.

Contribution

It proves that the sum of products of congruence classes equals a single class when gcd is 1, and that the sum of products of progressions eventually matches a specific progression.

Findings

Sum of products of classes equals a class when gcd is 1

Sum of products of progressions eventually matches a progression

Provides explicit formulas for these sets

Abstract

Consider the congruence class R_m(a)={a+im:i\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\in N_0}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m(b)+R_m(c)R_m(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+\ell m) for some i,j,k,\ell\in Z. It is proved that if gcd(a,b,c,d,m)=1, then R_m(a)R_m(b)+R_m(c)R_m(d) is equal to the congruence class R_m(ab+cd), and that the sum of products set P_m(a)P_m(b)+P_m(c)P_m(d) eventually coincides with the infinite arithmetic progression P_m(ab+cd).