Some results on pseudosquares

Abstract

If p p is an odd prime, the pseudosquare L p L_p is defined to be the least positive nonsquare integer such that L p ≡ 1 ( mod 8 ) L_p\equiv 1\pmod {8} and the Legendre symbol ( L p / q ) = 1 (L_p/q)=1 for all odd primes q ≤ p q\le p . In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L 271 L_{271} . We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that L p > e p / 2 L_p> e^{\sqrt {p/2}} , an inequality that must hold under the extended Riemann Hypothesis.