A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1

Michal Křížek; Jan Chleboun

Mathematica Bohemica (1994)

  • Volume: 119, Issue: 4, page 437-445
  • ISSN: 0862-7959

Abstract

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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 = ( k 2 n + 1 ) ( ℓ 2 n + 1 ) for some odd k and ℓ , k ≥ 3 , ℓ ≥ 3 , and integer n ≥ m + 2 , 3 n < 2 m . We prove that the greatest common divisor of k and ℓ is 1, k + ℓ ≡ 0 m o d 2 n , m a x ( k , ℓ ) ≥ F m - 2 , and either 3 | k or 3 | ℓ , i.e., 3 h 2 m + 2 + 1 | F m for an integer h ≥ 1 . Factorizations of F m into more than two factors are investigated as well. In particular, we prove that if F m = ( k 2 n + 1 ) 2 ( ℓ 2 j + 1 ) then j = n + 1 , 3 | ℓ and 5 | ℓ .

How to cite

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Křížek, Michal, and Chleboun, Jan. "A note on factorization of the Fermat numbers and their factors of the form $3h2^n+1$." Mathematica Bohemica 119.4 (1994): 437-445. <http://eudml.org/doc/29269>.

@article{Křížek1994,
abstract = {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 $F_m=(k2^n+1)(\ell 2^n+1)$ for some odd $k$ and $\ell , k\ge 3, \ell \ge 3$, and integer $n\ge m+2, 3n<2^m$. We prove that the greatest common divisor of $k$ and $\ell $ is 1, $k+\ell \equiv 0\ mod 2^n,\ max(k,\ell )\ge F_\{m-2\}$, and either $3|k$ or $3|\ell $, i.e., $3h2^\{m+2\}+1|F_m$ for an integer $h\ge 1$. Factorizations of $F_m$ into more than two factors are investigated as well. In particular, we prove that if $F_m=(k2^n+1)^2(\ell 2^j+1)$ then $j=n+1,3|\ell $ and $5|\ell $.},
author = {Křížek, Michal, Chleboun, Jan},
journal = {Mathematica Bohemica},
keywords = {congruence properties; Fermat numbers; prime numbers; factorization; squarefreensess; congruence properties},
language = {eng},
number = {4},
pages = {437-445},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on factorization of the Fermat numbers and their factors of the form $3h2^n+1$},
url = {http://eudml.org/doc/29269},
volume = {119},
year = {1994},
}

TY - JOUR
AU - Křížek, Michal
AU - Chleboun, Jan
TI - A note on factorization of the Fermat numbers and their factors of the form $3h2^n+1$
JO - Mathematica Bohemica
PY - 1994
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 119
IS - 4
SP - 437
EP - 445
AB - 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 $F_m=(k2^n+1)(\ell 2^n+1)$ for some odd $k$ and $\ell , k\ge 3, \ell \ge 3$, and integer $n\ge m+2, 3n<2^m$. We prove that the greatest common divisor of $k$ and $\ell $ is 1, $k+\ell \equiv 0\ mod 2^n,\ max(k,\ell )\ge F_{m-2}$, and either $3|k$ or $3|\ell $, i.e., $3h2^{m+2}+1|F_m$ for an integer $h\ge 1$. Factorizations of $F_m$ into more than two factors are investigated as well. In particular, we prove that if $F_m=(k2^n+1)^2(\ell 2^j+1)$ then $j=n+1,3|\ell $ and $5|\ell $.
LA - eng
KW - congruence properties; Fermat numbers; prime numbers; factorization; squarefreensess; congruence properties
UR - http://eudml.org/doc/29269
ER -

References

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  7. A. K. Lenstra H. W. Lenstra, Jr. M. S. Manasse J. M. Pollard, 10.1090/S0025-5718-1993-1182953-4, Math. Comp. 61 (1993), 319-349. (1993) MR1182953DOI10.1090/S0025-5718-1993-1182953-4
  8. M. A. Morrison J. Brillhart, A method of factoring and the factorization of F 7 , Math. Comp. 29 (1975), 183-205. (1975) MR0371800
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  10. H. C. Williams, How was F 6 factored?, Math. Comp. 61 (1993), 463-474. (1993) MR1182248

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