Report of a systematic review of Mersenne numbers 2^p-1 for p < 62982

Haworth, G. McC.

Holmes, S. M.

Hunt, D. J.

Lake, T. W.

Marron, P. J.

Reddaway, S. F.

Westbury, N.

English

Central Archive at the University of Reading

Abstracts of the American Mathematical Society, Vol. 4, No. 2, p. 196. ISSN 0192-5857:    83T-10-82  S M HOLMES, D J HUNT, T W LAKE, P J MARRON, S F REDDAWAY, N WESTBURY and G McC HAWORTH:            ICL, Reading, RG1 8PN, UK. Primality-testing Mersenne Numbers.  Preliminary Report. Mp = 2p-1, index p prime, is a Mersenne Number.  Let S1 = 4 and let Sn+1 = Sn2-2 mod Mp.  The Lucas-Lehmer primality test (LLT) is "Mp prime ⇔ residue Sp-1 = 0".  We have exercised the LLT on an index-set P using Fast Fermat-number-transform multiplication.  Codes A and B ran on an ICL DAP, an SIMD parallel processor having 4096 elementary processing elements.  Code C is under test.   P is a comprehensive set of primes for two reasons.  First, the code for the relatively complex transform algorithm deserves the fullest testing.  Secondly, the LLT is non-constructive and its history includes some incorrect residues which were temporarily thought to 'prove' their Mp composite. We believe an 'LLT proof' must include two independent residue computations.  Thus, while we did not include almost all p < 38220 like Nelson, P otherwise contains all p for which we knew or presumed the LLT had been applied.  P also contains all p < 62982 for which we knew of no Mp-factor. Our codes checked the squaring mod-7 and were run twice over their respective ranges.  Code A for p < 31488 tested M31487 in 142 seconds;  Code B for p < 62976 tested M62929 in 562 seconds. For some 2828 p less than the previous search-limit of 50024 we confirmed and filed a definitive set of residues.  We thus validated the results, as sometimes corrected, of Hurwitz/Selfridge, Kravitz/Berg, Gillies, Tuckerman, Nickel/Noll and Nelson/Slowinski.  For the range 50024 < p < 62982, our 475 new residues are all non-zero and so reveal no new prime Mp: we invite their confirmation.  (Received October 22, 1982) (Introduced by R. P. Brent)   

Primality-testing Mersenne numbers

Primality-testing Mersenne numbers

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