Reply-To: Richard E Crandall Sender: Number Theory List From: Richard E Crandall Subject: F_22 is composite Comments: To: NmbrThry@vm1.nodak.edu To: Multiple recipients of list NMBRTHRY F_22 was shown composite in October 1993 by R. Crandall, J. Doenias, C. Norrie, and J. Young. The run was done on an Amdahl mainframe and partly on a Cray. The multiply algorithm used the discrete weighted transform (DWT) of Crandall and B. Fagin, (to appear in Math. Comp.). The computation took about six Amdahl months. This may have been the longest computation ever performed for a non-trivial 1 bit result. The number of CPU ops spent was about 10^16. The group also verified independently the original F_20 compositeness result of Young and D. Buell (1988). For both of these numbers (and some other large Fermat cofactors also shown to be composite during these runs) some rigorous checking schemes were implemented. First, Norrie developed a "wavefront" scheme, in which the mainframe does the classical Pepin primality test (which does successive squaring) and a host of workstations follow the mainframe's results by checking squarings from the a-th to the b-th, for various small (b-a). In this way the deterministic character of the mainframe's Pepin run is verified. Norrie used a *different* program, in fact a different multiply algorithm, for the a-to-b verifications. All this time, Young, at Cray, verified various partial runs and attacked other numbers such as F_19, F_21 and their cofactors. All of these results will be put in print as soon as some final F_n verifications are performed; but F_22 is hereby announced as composite.