Divisors of Mersenne numbers
Author:
Samuel S. Wagstaff
Journal:
Math. Comp. 40 (1983), 385397
MSC:
Primary 10H15; Secondary 1004, 10A25
MathSciNet review:
679454
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Abstract 
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Abstract: We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbers in the range .
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 John Brillhart, "On the factors of certain Mersenne numbers, II," Math. Comp., v. 18, 1964, pp. 8792. MR 28 #2992. MR 0159776 (28:2992)
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 Cherwell, "Note on the distribution of the intervals between prime numbers," Quart. J. Math. Oxford, v. 17, 1946, pp. 4662. MR 8, 136. MR 0017308 (8:136e)
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 Cherwell & E. M. Wright, "The frequency of prime patterns," Quart. J. Math. Oxford (2), v. 11, 1960, pp. 6063. MR 24 #A98. MR 0130231 (24:A98)
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 John R. Ehrman, "The number of prime divisors of certain Mersenne numbers," Math. Comp., v. 21, 1967, pp. 700704. MR 36 #6368. MR 0223320 (36:6368)
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 Donald B. Gillies, "Three new Mersenne primes and a statistical theory," Math. Comp., v. 18, 1964, pp. 9397. MR 28 #2990. MR 0159774 (28:2990)
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 H. Halberstam & H. E. Richert, Sieve Methods, Academic Press, New York, 1974. MR 54 #12689. MR 0424730 (54:12689)
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 H. W. Lenstra, Jr., "Primality testing," Studieweek Getaltheorie en Computers, Sept. 15, 1980, Stichting Math. Centrum, Amsterdam.
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 Curt Noll & Laura Nickel, "The 25th and 26th Mersenne primes," Math. Comp., v. 35, 1980, pp. 13871390. MR 583517 (81k:10010)
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 G. Pólya, "Heuristic reasoning in the theory of numbers," Amer. Math. Monthly, v. 66, 1959, pp. 375384. MR 21 #3392. MR 0104639 (21:3392)
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 Daniel Shanks & Sidney Kravitz, "On the distribution of Mersenne divisors," Math. Comp., v. 21, 1967, pp. 97101. MR 36 #3717. MR 0220665 (36:3717)
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 David Slowinski, "Searching for the 27th Mersenne prime," J. Recreational Math., v. 11, 1979, pp. 258261. MR 80g: 10013. MR 536930 (80g:10013)
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 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, Fourth ed., Clarendon Press, London, 1960. MR 2445243 (2009i:11001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819830679454X
PII:
S 00255718(1983)0679454X
Keywords:
Mersenne number
Article copyright:
© Copyright 1983
American Mathematical Society
