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Some Results on Pseudosquares

Authors:	R F Lukes C D Patterson H C Williams

Institution:	Department of Computer Science, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 ; Xilinx Development Corporation, 52 Mortonhall Gate, Edinburgh EH16 6TJ, Scotland H. C. Williams ; Department of Computer Science, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Abstract:	If is an odd prime, the pseudosquare is defined to be the least positive nonsquare integer such that $L_p\equiv1\pmod{8}$ and the Legendre symbol for all odd primes $q\le p$ . 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 $L_{271}$ . We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that $L_p> e^{\sqrt{p/2}}$ , an inequality that must hold under the extended Riemann Hypothesis.

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