Divisors of Mersenne numbers

Author:
Samuel S. Wagstaff

Journal:
Math. Comp. **40** (1983), 385-397

MSC:
Primary 10H15; Secondary 10-04, 10A25

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679454-X

MathSciNet review:
679454

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Abstract | References | Similar Articles | Additional Information

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 .

**[1]**Paul T. Bateman and Roger A. Horn,*A heuristic asymptotic formula concerning the distribution of prime numbers*, Math. Comp.**16**(1962), 363–367. MR**0148632**, https://doi.org/10.1090/S0025-5718-1962-0148632-7**[2]**John Brillhart,*On the factors of certain Mersenne numbers. II*, Math. Comp.**18**(1964), 87–92. MR**0159776**, https://doi.org/10.1090/S0025-5718-1964-0159776-X**[3]**Cherwell,*Note on the distribution of the intervals between prime numbers*, Quart. J. Math., Oxford Ser.**17**(1946), 46–62. MR**0017308****[4]**Cherwell and E. M. Wright,*The frequency of prime-patterns*, Quart. J. Math. Oxford Ser. (2)**11**(1960), 60–63. MR**0130231**, https://doi.org/10.1093/qmath/11.1.60**[5]**John R. Ehrman,*The number of prime divisors of certain Mersenne numbers*, Math. Comp.**21**(1967), 700–704. MR**0223320**, https://doi.org/10.1090/S0025-5718-1967-0223320-1**[6]**Donald B. Gillies,*Three new Mersenne primes and a statistical theory*, Math. Comp.**18**(1964), 93–97. MR**0159774**, https://doi.org/10.1090/S0025-5718-1964-0159774-6**[7]**H. Halberstam and H.-E. Richert,*Sieve methods*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR**0424730****[8]**H. W. Lenstra, Jr., "Primality testing," Studieweek Getaltheorie en Computers, Sept. 1-5, 1980, Stichting Math. Centrum, Amsterdam.**[9]**Curt Noll and Laura Nickel,*The 25th and 26th Mersenne primes*, Math. Comp.**35**(1980), no. 152, 1387–1390. MR**583517**, https://doi.org/10.1090/S0025-5718-1980-0583517-4**[10]**G. Pólya,*Heuristic reasoning in the theory of numbers*, Amer. Math. Monthly**66**(1959), 375–384. MR**0104639**, https://doi.org/10.2307/2308748**[11]**Daniel Shanks and Sidney Kravitz,*On the distribution of Mersenne divisors*, Math. Comp.**21**(1967), 97–101. MR**0220665**, https://doi.org/10.1090/S0025-5718-1967-0220665-6**[12]**David Slowinski,*Searching for the 27th Mersenne prime*, J. Recreational Math.**11**(1978/79), no. 4, 258–267. MR**536930****[13]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR**2445243**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679454-X

Keywords:
Mersenne number

Article copyright:
© Copyright 1983
American Mathematical Society