Products of two random "primes'' in residue classesMPIM

by Bill Banks (University of Missouri, Columbia/MPIM)

Wednesday Sep 23, 2026, 2:30 PM → 3:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

Erdös, Odlyzko, and Sárközy conjectured that every invertible residue class mod $q$ contains a product of two primes of size at most $q$. This is open even under GRH. After reviewing random models of the primes (Cram\'er, Granville, and the Banks--Ford--Tao sieve model), I will show that the analogue of the EOS conjecture holds, in Granville's random model, almost surely and with both "primes" below $q^{3/4+\epsilon}$, along with a uniform asymptotic count. The arithmetic rests on Kloosterman sums; the probability requirement is minimal. Strikingly, the corresponding statement fails in the sieve model, so the choice of model is decisive.