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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:, robbiegranger@gmail.com
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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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References

1. Mersenne, M., Cogitata Physico Mathematica, Parisiis, 1644. Praefatio Generalis No. 19.Google Scholar
2. Uhler, H. S., On all of Mersenne's numbers particularly M193 , Nat. Acad. Sci., Proc., 34, (March 1948) pp. 102–103.CrossRefGoogle ScholarPubMed
3. Lucas, E., Nouveaux théorèrnes d'arithrnétique supérieure, C. R. A cad. Sci. Paris, 83 (1876) pp. 1286–1288.Google Scholar
4. Lehmer, D. H., On Lucas's test for the primality of Mersenne's numbers, J. London Math. Soc., 10 (1935) pp. 162–165.Google Scholar
5. Heath, T. L., A history of Greek mathematics, Vol. 2, Oxford: Clarendon Press (1921).Google Scholar
6. Bombelli, R., L 'A1gebra (1572).Google Scholar
7. Heath, T. L., The thirteen books of Euclid's Elements, Cambridge University Press (1908).Google Scholar
8. Fowler, D. and Robson, E., Square root approximations in old Babylonian mathematics: YBC 7289 in Context. Historia Mathematica 25 (1998) pp. 366–378.Google Scholar
9. Penrose, R., The road to reality: A complete guide to the laws of the universe, Vintage (2005).Google Scholar
10. Tao, T., The Lucas-Lehmer test for Mersenne primes. Available from http://terrytao.wordpress.com Google Scholar
11. Rosen, M. I., A proof of the Lucas-Lehmer test, The American Mathematical Monthly, 95 (1988) pp. 855–856.Google Scholar
12. Bruce, J. W., A really trivial proof of the Lucas-Lehmer test, The American Mathematical Monthly, 100 (1993) pp. 370–371.Google Scholar
13. Knuth, D. E., The art of computer programming, v. 2. Seminumerical Algorithms (2nd edn.), Addison-Wesley (1981).Google Scholar
14. Stehle, D. and Zimmermann, P., A binary recursive gcd algorithm, Algorithmic Number Theory (ANTS-VI), LNCS 3076, Springer (2004) pp.411–425.Google Scholar
15. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (6th edn.) Oxford University Press (2008).CrossRefGoogle Scholar
16. Herstein, I. N., Abstract algebra (3rd edn.), John Wiley & Sons (1996).Google Scholar
17. Crandall, R. E. and Pomerance, C., Prime numbers: a computational perspective (2nd edn.), Springer, New York (2005).Google Scholar
18. Gross, B. H., An elliptic curve test for Mersenne primes, J. Number Theory, 110 (2005) pp. 114–119.CrossRefGoogle Scholar
19. Lemmerrneyer, F., Conics – a poor man's elliptic curves, (2003). Available from: http://arxiv.org/pdf/math/0311306v1.pdf Google Scholar
20. Lemmerrneyer, F., Kreise und Quadrate modulo p, Mathem. Semesterberichte 47 (2000) pp. 51–73.Google Scholar
21. Wagstaff, S. S. Jr, Divisors of Mersenne numbers, Math. Comp., 40 (1983) pp. 385–397.Google Scholar