On smooth square-free numbers in arithmetic progressions

Booker and Pomerance [Proc. Amer. Math. Soc. 145 (2017) 5035-5042] have shown that any residue class modulo a prime p > 11 can be represented by a positive p-smooth square-free integer s=pO(logp) with all prime factors up to p and conjectured that in fact one can find such s with s=pO(1). Using bounds on double Kloosterman sums due to Garaev [Mat. Zametki 88 (2010) 365-373] we prove this conjecture in a stronger form s <= p3/2+o(1) and also consider more general versions of this question replacing p-smoothness of s by the stronger condition of p alpha-smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer s <= p2+o(1) which is p1/(4e1/2)+o(1)-smooth. Additionally, we obtain stronger results for almost all primes p.