Doubly-focused enumeration of pseudosquares and pseudocubes

Abstract

This paper offers numerical evidence for a conjecture that primality proving may be done in (log N)3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences - doubly-focused enumeration - is examined. This technique, first described by D. J. Bernstein, allowed us to obtain record-setting sieve computations in software on general purpose computers. © Springer-Verlag Berlin Heidelberg 2006.