Factors of Fermat numbers and large primes of the form

Author:
Wilfrid Keller

Journal:
Math. Comp. **41** (1983), 661-673

MSC:
Primary 11Y05; Secondary 11A41, 11Y11

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717710-7

MathSciNet review:
717710

Full-text 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. A-373.**[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. 1333-1336. 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. 229-231. 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. 253-255. 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. 627-630. 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. 1419-1421. 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. 14-19.**[10]**Gary B. Gostin & Philip B. McLaughlin, Jr., "Six new factors of Fermat numbers,"*Math. Comp.*, v. 38, 1982, pp. 645-649. MR**645680 (83c:10003)****[11]**Richard K. Guy,*Unsolved Problems in Number Theory*, Springer-Verlag, 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. 109-112. 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. 381-384. MR**679453 (84k:10006)****[15]**Donald E. Knuth,*The Art of Computer Programming*, Vol. 2:*Seminumerical Algorithms*, 2nd ed., Addison-Wesley, Reading, Mass., 1981. MR**633878 (83i:68003)****[16]**D. H. Lehmer, "On Fermat's quotient, base two,"*Math. Comp.*, v. 36, 1981, pp. 289-290. MR**595064 (82e:10004)****[17]**G. Matthew & H. C. Williams, "Some new primes of the form ,"*Math. Comp.*, v. 31, 1977, pp. 797-798. 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. 183-205. MR**51**#8017. MR**0371800 (51:8017)****[19]**P. Ribenboim, "On the square factors of the numbers of Fermat and Ferentinou-Nicolacopoulou,"*Bull. Soc. Math. Grèce (N. S.)*, v. 20, 1979, pp. 81-92. 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. 673-681. MR**20**#3097. MR**0096614 (20:3097)****[21]**W. Sierpiński, "Sur un problème concernant les nombres ,"*Elem. Math.*, v. 15, 1960, pp. 73-74. 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, Springer-Verlag, Berlin and Heidelberg, 1981, pp. 107-114. MR**641240 (84j:10009)****[23]**Hiromi Suyama, "Searching for prime factors of Fermat numbers with a microcomputer,"*bit*, v. 13, 1981, pp. 240-245. (Japanese) MR**82c**: 10012. MR**610300 (82c:10012)****[24]**H. C. Williams, "Primality testing on a computer,"*Ars Combin.*, v. 5, 1978, pp. 127-185. MR**80d**: 10002. MR**504864 (80d:10002)**

Retrieve articles in *Mathematics of Computation*
with MSC:
11Y05,
11A41,
11Y11

Retrieve articles in all journals with MSC: 11Y05, 11A41, 11Y11

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717710-7

Keywords:
Fermat numbers,
numbers of Ferentinou-Nicolacopoulou,
factoring,
trial division,
large primes,
covering set of divisors,
twin primes

Article copyright:
© Copyright 1983
American Mathematical Society