| Title:
|
A note on factorization of the Fermat numbers and their factors of the form $3h2\sp n+1$ (English) |
| Author:
|
Křížek, Michal |
| Author:
|
Chleboun, Jan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
119 |
| Issue:
|
4 |
| Year:
|
1994 |
| Pages:
|
437-445 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that any factorization of any composite Fermat number $F_m={2^{2}}^m+1$ into two nontrivial factors can be expressed in the form $F_m=(k2^n+1)(\ell2^n+1)$ for some odd $k$ and $\ell, k\geq 3, \ell \geq 3$, and integer $n\geq m+2, 3n<2^m$. We prove that the greatest common divisor of $k$ and $\ell$ is 1, $k+\ell\equiv 0\ mod 2^n,\ max(k,\ell)\geq F_{m-2}$, and either $3|k$ or $3|\ell$, i.e., $3h2^{m+2}+1|F_m$ for an integer $h\geq 1$. Factorizations of $F_m$ into more than two factors are investigated as well. In particular, we prove that if $F_m=(k2^n+1)^2(\ell2^j+1)$ then $j=n+1,3|\ell$ and $5|\ell$. (English) |
| Keyword:
|
congruence properties |
| Keyword:
|
Fermat numbers |
| Keyword:
|
prime numbers |
| Keyword:
|
factorization |
| Keyword:
|
squarefreensess |
| MSC:
|
11A51 |
| MSC:
|
11Y05 |
| idZBL:
|
Zbl 0822.11007 |
| idMR:
|
MR1316595 |
| DOI:
|
10.21136/MB.1994.126115 |
| . |
| Date available:
|
2009-09-24T21:08:00Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126115 |
| . |
| Reference:
|
[1] R. P. Brent: Factorization of the eleventh Fermat number.Abstracts Amer. Math. Soc. 10 (1989), 176-177. |
| Reference:
|
[2] R. P. Brent J. M. Pollard: Factorization of the eight Fermat number.Math. Comp. 36 (1981), 627-630. MR 0606520, 10.1090/S0025-5718-1981-0606520-5 |
| Reference:
|
[3] J. Brillhart D. H. Lehmer J. L. Selfridge B. Tuckerman S. S. Wagstaff: Factorization of $b^n \pm 1$, b = 2,3,5,6,7,10,11,12 up to high powers.Contemporary Math. vol. 22, Amer. Math. Soc., Providence, 1988. MR 0715603 |
| Reference:
|
[4] L. E. Dickson: History of the theory of numbers, vol. I, Divisibility and primality.Carnegie Inst., Washington, 1919. |
| Reference:
|
[5] G. H. Hardy E. M. Wright: An introduction to the theory of numbers.Clarendon Press, Oxford, 1945. |
| Reference:
|
[6] W. Keller: Factors of Fermat numbers and large primes of the form $k 2^n +1$. II.Preprint Univ. of Hamburg, 1992, 1-40. MR 0717710 |
| Reference:
|
[7] A. K. Lenstra H. W. Lenstra, Jr. M. S. Manasse J. M. Pollard: The factorization of the ninth Fermat number.Math. Comp. 61 (1993), 319-349. MR 1182953, 10.1090/S0025-5718-1993-1182953-4 |
| Reference:
|
[8] M. A. Morrison J. Brillhart: A method of factoring and the factorization of $F_7$.Math. Comp. 29 (1975), 183-205. MR 0371800 |
| Reference:
|
[9] N. Robbins: Beginning number theory.W. C. Brown Publishers, 1993. Zbl 0824.11001 |
| Reference:
|
[10] H. C. Williams: How was $F_6$ factored?.Math. Comp. 61 (1993), 463-474. MR 1182248 |
| . |
