# A note on factorization of the Fermat numbers and their factors of the form $3h{2}^{n}+1$

Mathematica Bohemica (1994)

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

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topKřížek, 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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