Products in Residue Classes

We consider a problem of P.~Erd{\H o}s, A.~M.~Odlyzko and A.~S{á}rk{\H o}zy about the representation of residue classes modulo $m$ by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results “on average” over moduli $m$ and residue classes modulo $m$ and somewhat stronger results when the average is restricted to prime moduli $m = p$. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.

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Published 1 January 2008