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Keywords:
congruence properties; Fermat numbers; prime numbers; factorization; squarefreensess
congruence properties; Fermat numbers; prime numbers; factorization; squarefreensess
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 $5|\ell$.
References:
[1] R. P. Brent: Factorization of the eleventh Fermat number. Abstracts Amer. Math. Soc. 10 (1989), 176-177.
[2] R. P. Brent J. M. Pollard: Factorization of the eight Fermat number. Math. Comp. 36 (1981), 627-630. DOI 10.1090/S0025-5718-1981-0606520-5 | MR 0606520
[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
[4] L. E. Dickson: History of the theory of numbers, vol. I, Divisibility and primality. Carnegie Inst., Washington, 1919.
[5] G. H. Hardy E. M. Wright: An introduction to the theory of numbers. Clarendon Press, Oxford, 1945.
[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
[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. DOI 10.1090/S0025-5718-1993-1182953-4 | MR 1182953
[8] M. A. Morrison J. Brillhart: A method of factoring and the factorization of $F_7$. Math. Comp. 29 (1975), 183-205. MR 0371800
[9] N. Robbins: Beginning number theory. W. C. Brown Publishers, 1993. Zbl 0824.11001
[10] H. C. Williams: How was $F_6$ factored?. Math. Comp. 61 (1993), 463-474. MR 1182248
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