Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ 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 . We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp > e √ p/2 , an inequality that must hold under the extended Riemann Hypothesis.

Abstract. Let p ( D ) be the period length of the continued fraction for fD . Under the extended Riemann Hypothesis for S(fD ) one would expect that p(D) = 0(D1/2 log log D). In order to test this it is necessary to find values of D for which p( D) is large. This, in turn, requires that we be able to find solutions to large sets of simultaneous linear congruences. The