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Could, or should, the ancient Greeks have discovered the Lucas-Lehmer test?

Published online by Cambridge University Press:  23 January 2015

Robert Granger
Robert Granger*
Affiliation:
School of Mathematical Sciences, University College Dublin, Irelande-mail:, [email protected]
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Extract

The Lucas-Lehmer (LL) test is the most efficient known for testing the primality of Mersenne numbers, i.e. the integers Ml = 2l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range, namely l = 2,3,5,7, 13, 17, 19,31,67, 127 and 257 [1]. Mersenne's list turned out to be incorrect, omitting the prime-producing l = 61, 89 and 107 and including the composite-producing l = 67 and 257, although this was not finally confirmed until 1947, using both the LL test and contemporary mechanical calculators [2]. The LL test is based on the following theorem.

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 539 , July 2013 , pp. 242 - 255
Copyright
Copyright © The Mathematical Association 2013

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