Divisors of Mersenne numbers
Author:
Samuel S. Wagstaff
Journal:
Math. Comp. 40 (1983), 385397
MSC:
Primary 10H15; Secondary 1004, 10A25
MathSciNet review:
679454
Fulltext 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 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
(26 #6139), http://dx.doi.org/10.1090/S00255718196201486327
 [2]
John
Brillhart, On the factors of certain Mersenne
numbers. II, Math. Comp. 18 (1964), 87–92. MR 0159776
(28 #2992), http://dx.doi.org/10.1090/S0025571819640159776X
 [3]
Cherwell,
Note on the distribution of the intervals between prime numbers,
Quart. J. Math., Oxford Ser. 17 (1946), 46–62. MR 0017308
(8,136e)
 [4]
Cherwell
and E.
M. Wright, The frequency of primepatterns, Quart. J. Math.
Oxford Ser. (2) 11 (1960), 60–63. MR 0130231
(24 #A98)
 [5]
John
R. Ehrman, The number of prime divisors of
certain Mersenne numbers, Math. Comp. 21 (1967), 700–704.
MR
0223320 (36 #6368), http://dx.doi.org/10.1090/S00255718196702233201
 [6]
Donald
B. Gillies, Three new Mersenne primes and a
statistical theory, Math. Comp. 18 (1964), 93–97. MR 0159774
(28 #2990), http://dx.doi.org/10.1090/S00255718196401597746
 [7]
H.
Halberstam and H.E.
Richert, Sieve methods, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], LondonNew York, 1974. London
Mathematical Society Monographs, No. 4. MR 0424730
(54 #12689)
 [8]
H. W. Lenstra, Jr., "Primality testing," Studieweek Getaltheorie en Computers, Sept. 15, 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
(81k:10010), http://dx.doi.org/10.1090/S00255718198005835174
 [10]
G.
Pólya, Heuristic reasoning in the theory of numbers,
Amer. Math. Monthly 66 (1959), 375–384. MR 0104639
(21 #3392)
 [11]
Daniel
Shanks and Sidney
Kravitz, On the distribution of Mersenne
divisors, Math. Comp. 21 (1967), 97–101. MR 0220665
(36 #3717), http://dx.doi.org/10.1090/S00255718196702206656
 [12]
David
Slowinski, Searching for the 27th Mersenne prime, J.
Recreational Math. 11 (1978/79), no. 4,
258–267. MR
536930 (80g:10013)
 [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. HeathBrown and J.
H. Silverman; With a foreword by Andrew Wiles. MR 2445243
(2009i:11001)
 [1]
 Paul T. Bateman & Roger A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers," Math. Comp., v. 16, 1962, pp. 363367. MR 26 #6139. MR 0148632 (26:6139)
 [2]
 John Brillhart, "On the factors of certain Mersenne numbers, II," Math. Comp., v. 18, 1964, pp. 8792. MR 28 #2992. MR 0159776 (28:2992)
 [3]
 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)
 [4]
 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)
 [5]
 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)
 [6]
 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)
 [7]
 H. Halberstam & H. E. Richert, Sieve Methods, Academic Press, New York, 1974. MR 54 #12689. MR 0424730 (54:12689)
 [8]
 H. W. Lenstra, Jr., "Primality testing," Studieweek Getaltheorie en Computers, Sept. 15, 1980, Stichting Math. Centrum, Amsterdam.
 [9]
 Curt Noll & Laura Nickel, "The 25th and 26th Mersenne primes," Math. Comp., v. 35, 1980, pp. 13871390. MR 583517 (81k:10010)
 [10]
 G. Pólya, "Heuristic reasoning in the theory of numbers," Amer. Math. Monthly, v. 66, 1959, pp. 375384. MR 21 #3392. MR 0104639 (21:3392)
 [11]
 Daniel Shanks & Sidney Kravitz, "On the distribution of Mersenne divisors," Math. Comp., v. 21, 1967, pp. 97101. MR 36 #3717. MR 0220665 (36:3717)
 [12]
 David Slowinski, "Searching for the 27th Mersenne prime," J. Recreational Math., v. 11, 1979, pp. 258261. MR 80g: 10013. MR 536930 (80g:10013)
 [13]
 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, Fourth ed., Clarendon Press, London, 1960. MR 2445243 (2009i:11001)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
10H15,
1004,
10A25
Retrieve articles in all journals
with MSC:
10H15,
1004,
10A25
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
