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arXiv · 0707.3476

Sums of products of congruence classes and of arithmetic progressions

Sergei V. Konyagin·Melvyn B. Nathanson
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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).

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Sergei V. Konyagin, Melvyn B. Nathanson. 2007-07-24. Sums of products of congruence classes and of arithmetic progressions. https://arxiv.org/abs/0707.3476

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