Factors of Fermat numbers and large primes of the form
Author:
Wilfrid Keller
Journal:
Math. Comp. 41 (1983), 661673
MSC:
Primary 11Y05; Secondary 11A41, 11Y11
MathSciNet review:
717710
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A new factor is given for each of the Fermat numbers , and . In addition, a factor of discovered by Gary Gostin is presented. The current status for all is shown in a table. Primes of the form odd, are listed for , , and for , . Some primes for even larger values of n are included, the largest one being . Also, a survey of several related questions is given. In particular, values of k such that is composite for every n are considered, as well as odd values of h such that never yields a twin prime pair.
 [1]
A. O. L. Atkin & N. W. Rickert, "On a larger pair of twin primes," Notices Amer. Math. Soc., v. 26, 1979, p. A373.
 [2]
A. O. L. Atkin & N. W. Rickert, "Some factors of Fermat numbers," Abstracts Amer. Math. Soc., v. 1, 1980, p. 211.
 [3]
Robert
Baillie, New primes of the form
𝑘⋅2ⁿ+1, Math. Comp.
33 (1979), no. 148, 1333–1336. MR 537979
(80h:10009), http://dx.doi.org/10.1090/S00255718197905379790
 [4]
Robert
Baillie, G.
Cormack, and H.
C. Williams, The problem of Sierpiński
concerning 𝑘⋅2ⁿ+1, Math.
Comp. 37 (1981), no. 155, 229–231. MR 616376
(83a:10006a), http://dx.doi.org/10.1090/S00255718198106163762
 [5]
Richard
P. Brent, Succinct proofs of primality for the
factors of some Fermat numbers, Math. Comp.
38 (1982), no. 157, 253–255. MR 637304
(82k:10002), http://dx.doi.org/10.1090/S00255718198206373040
 [6]
Richard
P. Brent and John
M. Pollard, Factorization of the eighth Fermat
number, Math. Comp. 36
(1981), no. 154, 627–630. MR 606520
(83h:10014), http://dx.doi.org/10.1090/S00255718198106065205
 [7]
Ingo Büchel & Wilfrid Keller, Ein Programmsystem für Rationale Arithmetik: Einführung und Beispielsammlung, Bericht Nr. 8004, Rechenzentrum der Universität Hamburg, April 1980.
 [8]
G.
V. Cormack and H.
C. Williams, Some very large primes of the form
𝑘⋅2^{𝑚}+1, Math. Comp.
35 (1980), no. 152, 1419–1421. MR 583519
(81i:10011), http://dx.doi.org/10.1090/S00255718198005835198
 [9]
Martin Gardner, "Mathematical games: Gauss's congruence theory was mod as early as 1801," Scientific American, v. 244, #2, February 1981, pp. 1419.
 [10]
Gary
B. Gostin and Philip
B. McLaughlin Jr., Six new factors of Fermat
numbers, Math. Comp. 38
(1982), no. 158, 645–649. MR 645680
(83c:10003), http://dx.doi.org/10.1090/S00255718198206456808
 [11]
Richard
K. Guy, Unsolved problems in number theory, Unsolved Problems
in Intuitive Mathematics, vol. 1, SpringerVerlag, New York, 1981.
Problem Books in Mathematics. MR 656313
(83k:10002)
 [12]
John
C. Hallyburton Jr. and John
Brillhart, Two new factors of Fermat
numbers, Math. Comp. 29 (1975), 109–112.
Collection of articles dedicated to Derrick Henry Lehmer on the occasion of
his seventieth birthday. MR 0369225
(51 #5460), http://dx.doi.org/10.1090/S00255718197503692251
 [13]
G.
H. Hardy and E.
M. Wright, An introduction to the theory of numbers, 5th ed.,
The Clarendon Press Oxford University Press, New York, 1979. MR 568909
(81i:10002)
 [14]
G.
Jaeschke, On the smallest 𝑘 such that
all 𝑘⋅2ⁿ+1 are composite, Math. Comp. 40 (1983), no. 161, 381–384. MR 679453
(84k:10006), http://dx.doi.org/10.1090/S00255718198306794538
 [15]
Donald
E. Knuth, The art of computer programming. Vol. 2, 2nd ed.,
AddisonWesley Publishing Co., Reading, Mass., 1981. Seminumerical
algorithms; AddisonWesley Series in Computer Science and Information
Processing. MR
633878 (83i:68003)
 [16]
D.
H. Lehmer, On Fermat’s quotient, base
two, Math. Comp. 36
(1981), no. 153, 289–290. MR 595064
(82e:10004), http://dx.doi.org/10.1090/S00255718198105950645
 [17]
G.
Matthew and H.
C. Williams, Some new primes of the form
𝑘⋅2ⁿ+1, Math. Comp.
31 (1977), no. 139, 797–798. MR 0439719
(55 #12605), http://dx.doi.org/10.1090/S00255718197704397190
 [18]
Michael
A. Morrison and John
Brillhart, A method of factoring and the
factorization of 𝐹₇, Math.
Comp. 29 (1975),
183–205. Collection of articles dedicated to Derrick Henry Lehmer on
the occasion of his seventieth birthday. MR 0371800
(51 #8017), http://dx.doi.org/10.1090/S00255718197503718005
 [19]
P.
Ribenboim, On the square factors of the numbers of Fermat and
FerentinouNicolacopoulou, Bull. Soc. Math. Grèce (N.S.)
20 (1979), 81–92. MR 642432
(83f:10016)
 [20]
Raphael
M. Robinson, A report on primes of the form
𝑘⋅2ⁿ+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc. 9 (1958), 673–681. MR 0096614
(20 #3097), http://dx.doi.org/10.1090/S00029939195800966147
 [21]
W.
Sierpiński, Sur un problème concernant les nombres
𝑘⋅2ⁿ+1, Elem. Math. 15 (1960),
73–74 (French). MR 0117201
(22 #7983)
 [22]
R.
G. Stanton, Further results on covering integers of the form
1+𝑘2ⁿ by primes, Combinatorial mathematics, VIII
(Geelong, 1980) Lecture Notes in Math., vol. 884, Springer, Berlin,
1981, pp. 107–114. MR 641240
(84j:10009)
 [23]
Hiromi
Suyama, Searching for prime factors of Fermat numbers with a
microcomputer, BIT (Tokyo) 13 (1981), no. 3,
240–245 (Japanese). MR 610300
(82c:10012)
 [24]
H.
C. Williams, Primality testing on a computer, Ars Combin.
5 (1978), 127–185. MR 504864
(80d:10002)
 [1]
 A. O. L. Atkin & N. W. Rickert, "On a larger pair of twin primes," Notices Amer. Math. Soc., v. 26, 1979, p. A373.
 [2]
 A. O. L. Atkin & N. W. Rickert, "Some factors of Fermat numbers," Abstracts Amer. Math. Soc., v. 1, 1980, p. 211.
 [3]
 Robert Baillie, "New primes of the form ," Math. Comp., v. 33, 1979, pp. 13331336. MR 80h: 10009. Table errata: MTE 585, Math. Comp., v. 38, 1982, p. 335. MR 82m: 10013. MR 537979 (80h:10009)
 [4]
 Robert Baillie, G. Cormack & H. C. Williams, "The problem of Sierpiński concerning ," Math. Comp., v. 37, 1981, pp. 229231. MR 83a: 10006a. Corrigenda: Math. Comp., v. 39, 1982, p. 308. MR 83a: 10006b. MR 616376 (83a:10006a)
 [5]
 Richard P. Brent, "Succinct proofs of primality for the factors of some Fermat numbers," Math. Comp., v. 38, 1982, pp. 253255. MR 82k: 10002. MR 637304 (82k:10002)
 [6]
 Richard P. Brent & John M. Pollard, "Factorization of the eighth Fermat number," Math. Comp., v. 36, 1981, pp. 627630. MR 606520 (83h:10014)
 [7]
 Ingo Büchel & Wilfrid Keller, Ein Programmsystem für Rationale Arithmetik: Einführung und Beispielsammlung, Bericht Nr. 8004, Rechenzentrum der Universität Hamburg, April 1980.
 [8]
 G. V. Cormack & H. C. Williams, "Some very large primes of the form " Math. Comp., v. 35, 1980, pp. 14191421. MR 81i: 10011. Table Errata: MTE 586, Math. Comp., v. 38, 1982, p. 335. MR 82k: 10011. MR 583519 (81i:10011)
 [9]
 Martin Gardner, "Mathematical games: Gauss's congruence theory was mod as early as 1801," Scientific American, v. 244, #2, February 1981, pp. 1419.
 [10]
 Gary B. Gostin & Philip B. McLaughlin, Jr., "Six new factors of Fermat numbers," Math. Comp., v. 38, 1982, pp. 645649. MR 645680 (83c:10003)
 [11]
 Richard K. Guy, Unsolved Problems in Number Theory, SpringerVerlag, New York, 1981. MR 656313 (83k:10002)
 [12]
 John C. Hallyburton, Jr & John Brillhart, "Two new factors of Fermat numbers," Math. Comp., v. 29, 1975, pp. 109112. MR 51 #5460. Corrigendum: Math. Comp., v. 30, 1976, p. 198. MR 52 #13599. MR 0369225 (51:5460)
 [13]
 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford, 1979. MR 568909 (81i:10002)
 [14]
 G. Jaeschke, "On the smallest k such that all are composite," Math. Comp., v. 40, 1983, pp. 381384. MR 679453 (84k:10006)
 [15]
 Donald E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed., AddisonWesley, Reading, Mass., 1981. MR 633878 (83i:68003)
 [16]
 D. H. Lehmer, "On Fermat's quotient, base two," Math. Comp., v. 36, 1981, pp. 289290. MR 595064 (82e:10004)
 [17]
 G. Matthew & H. C. Williams, "Some new primes of the form ," Math. Comp., v. 31, 1977, pp. 797798. MR 55 #12605. MR 0439719 (55:12605)
 [18]
 Michael A. Morrison & John Brillhart, "A method of factoring and the factorization of ," Math. Comp., v. 29, 1975, pp. 183205. MR 51 #8017. MR 0371800 (51:8017)
 [19]
 P. Ribenboim, "On the square factors of the numbers of Fermat and FerentinouNicolacopoulou," Bull. Soc. Math. Grèce (N. S.), v. 20, 1979, pp. 8192. MR 642432 (83f:10016)
 [20]
 Raphael M. Robinson, "A report on primes of the form and on factors of Fermat numbers," Proc. Amer. Math. Soc., v. 9, 1958, pp. 673681. MR 20 #3097. MR 0096614 (20:3097)
 [21]
 W. Sierpiński, "Sur un problème concernant les nombres ," Elem. Math., v. 15, 1960, pp. 7374. MR 22 #7983. Corrigendum: Elem. Math., v. 17, 1962, p. 85. MR 0117201 (22:7983)
 [22]
 R. G. Stanton, "Further results on covering integers of the form by primes," Lecture Notes in Math., Vol. 884: Combinatorial Mathematics VIII, SpringerVerlag, Berlin and Heidelberg, 1981, pp. 107114. MR 641240 (84j:10009)
 [23]
 Hiromi Suyama, "Searching for prime factors of Fermat numbers with a microcomputer," bit, v. 13, 1981, pp. 240245. (Japanese) MR 82c: 10012. MR 610300 (82c:10012)
 [24]
 H. C. Williams, "Primality testing on a computer," Ars Combin., v. 5, 1978, pp. 127185. MR 80d: 10002. MR 504864 (80d:10002)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
11Y05,
11A41,
11Y11
Retrieve articles in all journals
with MSC:
11Y05,
11A41,
11Y11
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307177107
PII:
S 00255718(1983)07177107
Keywords:
Fermat numbers,
numbers of FerentinouNicolacopoulou,
factoring,
trial division,
large primes,
covering set of divisors,
twin primes
Article copyright:
© Copyright 1983 American Mathematical Society
