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

MathSciNet review:
717710

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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 𝑘⋅2ⁿ+1*, Math. Comp.**33**(1979), no. 148, 1333–1336. MR**537979**, 10.1090/S0025-5718-1979-0537979-0**[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**, 10.1090/S0025-5718-1981-0616376-2**[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**, 10.1090/S0025-5718-1982-0637304-0**[6]**Richard P. Brent and John M. Pollard,*Factorization of the eighth Fermat number*, Math. Comp.**36**(1981), no. 154, 627–630. MR**606520**, 10.1090/S0025-5718-1981-0606520-5**[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**, 10.1090/S0025-5718-1980-0583519-8**[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 and Philip B. McLaughlin Jr.,*Six new factors of Fermat numbers*, Math. Comp.**38**(1982), no. 158, 645–649. MR**645680**, 10.1090/S0025-5718-1982-0645680-8**[11]**Richard K. Guy,*Unsolved problems in number theory*, Unsolved Problems in Intuitive Mathematics, vol. 1, Springer-Verlag, New York-Berlin, 1981. Problem Books in Mathematics. MR**656313****[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**, 10.1090/S0025-5718-1975-0369225-1**[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****[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**, 10.1090/S0025-5718-1983-0679453-8**[15]**Donald E. Knuth,*The art of computer programming. Vol. 2*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**633878****[16]**D. H. Lehmer,*On Fermat’s quotient, base two*, Math. Comp.**36**(1981), no. 153, 289–290. MR**595064**, 10.1090/S0025-5718-1981-0595064-5**[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**, 10.1090/S0025-5718-1977-0439719-0**[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**, 10.1090/S0025-5718-1975-0371800-5**[19]**P. Ribenboim,*On the square factors of the numbers of Fermat and Ferentinou-Nicolacopoulou*, Bull. Soc. Math. Grèce (N.S.)**20**(1979), 81–92. MR**642432****[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**, 10.1090/S0002-9939-1958-0096614-7**[21]**W. Sierpiński,*Sur un problème concernant les nombres 𝑘⋅2ⁿ+1*, Elem. Math.**15**(1960), 73–74 (French). MR**0117201****[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-New York, 1981, pp. 107–114. MR**641240****[23]**Hiromi Suyama,*Searching for prime factors of Fermat numbers with a microcomputer*, BIT (Tokyo)**13**(1981), no. 3, 240–245 (Japanese). MR**610300****[24]**H. C. Williams,*Primality testing on a computer*, Ars Combin.**5**(1978), 127–185. MR**504864**

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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