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

Full-text PDF Free Access

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 ${M_p}$ in the range $17000 < p < {10^5}$.

- Paul T. Bateman and Roger A. Horn,
*A heuristic asymptotic formula concerning the distribution of prime numbers*, Math. Comp.**16**(1962), 363–367. MR**148632**, DOI https://doi.org/10.1090/S0025-5718-1962-0148632-7 - John Brillhart,
*On the factors of certain Mersenne numbers. II*, Math. Comp.**18**(1964), 87–92. MR**159776**, DOI https://doi.org/10.1090/S0025-5718-1964-0159776-X - Cherwell,
*Note on the distribution of the intervals between prime numbers*, Quart. J. Math. Oxford Ser.**17**(1946), 46–62. MR**17308**, DOI https://doi.org/10.1093/qmath/os-17.1.46 - Cherwell and E. M. Wright,
*The frequency of prime-patterns*, Quart. J. Math. Oxford Ser. (2)**11**(1960), 60–63. MR**130231**, DOI https://doi.org/10.1093/qmath/11.1.60 - John R. Ehrman,
*The number of prime divisors of certain Mersenne numbers*, Math. Comp.**21**(1967), 700–704. MR**223320**, DOI https://doi.org/10.1090/S0025-5718-1967-0223320-1 - Donald B. Gillies,
*Three new Mersenne primes and a statistical theory*, Math. Comp.**18**(1964), 93–97. MR**159774**, DOI https://doi.org/10.1090/S0025-5718-1964-0159774-6 - 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**
H. W. Lenstra, Jr., "Primality testing," Studieweek Getaltheorie en Computers, Sept. 1-5, 1980, Stichting Math. Centrum, Amsterdam.
- Curt Noll and Laura Nickel,
*The 25th and 26th Mersenne primes*, Math. Comp.**35**(1980), no. 152, 1387–1390. MR**583517**, DOI https://doi.org/10.1090/S0025-5718-1980-0583517-4 - G. Pólya,
*Heuristic reasoning in the theory of numbers*, Amer. Math. Monthly**66**(1959), 375–384. MR**104639**, DOI https://doi.org/10.2307/2308748 - Daniel Shanks and Sidney Kravitz,
*On the distribution of Mersenne divisors*, Math. Comp.**21**(1967), 97–101. MR**220665**, DOI https://doi.org/10.1090/S0025-5718-1967-0220665-6 - David Slowinski,
*Searching for the 27th Mersenne prime*, J. Recreational Math.**11**(1978/79), no. 4, 258–267. MR**536930** - 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**

Retrieve articles in *Mathematics of Computation*
with MSC:
10H15,
10-04,
10A25

Retrieve articles in all journals with MSC: 10H15, 10-04, 10A25

Additional Information

Keywords:
Mersenne number

Article copyright:
© Copyright 1983
American Mathematical Society