On the smallest simultaneous power nonresidue modulo a prime

Abstract Let p be a prime and let p 1 , … , p r ${p_{1},\ldots,p_{r}}$ be distinct prime divisors of p - 1 ${p-1}$ . We prove that the smallest positive integer n which is a simultaneous p 1 , … , p r ${p_{1},\ldots,p_{r}}$ -power nonresidue modulo p satisfies n < p 1 / 4 - c r + o ⁢ ( 1 ) ( p → ∞ ) $n for some positive c r ${c_{r}}$ satisfying c r = e - ( 1 + o ⁢ ( 1 ) ) ⁢ r ${c_{r}=e^{-(1+o(1))r}}$ as r → ∞ ${r\to\infty}$ .