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Mathematics of Computation

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Factors of generalized Fermat numbers

Authors: Anders Björn and Hans Riesel
Journal: Math. Comp. 67 (1998), 441-446
MSC (1991): Primary 11-04, 11A51, 11Y05, 11Y11
Erratum: Math. Comp. 74 (2005), 2099
Erratum: Math. Comp. 80 (2011), 1865-1866
Supplement: Additional information related to this article.
MathSciNet review: 1433262
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Abstract | References | Similar Articles | Additional Information

Abstract: A search for prime factors of the generalized Fermat numbers $F_n(a,b)=a^{2^n}+b^{2^n}$ has been carried out for all pairs $(a,b)$ with $a,b\leq 12$ and GCD$(a,b)=1$. The search limit $k$ on the factors, which all have the form $p=k\cdot 2^m+1$, was $k=10^9$ for $m\leq 100$ and $k=3\cdot 10^6$ for $101\leq m\leq 1000$. Many larger primes of this form have also been tried as factors of $F_n(a,b)$. Several thousand new factors were found, which are given in our tables.-For the smaller of the numbers, i.e. for $n\leq 15$, or, if $a,b\leq 8$, for $n\leq 16$, the cofactors, after removal of the factors found, were subjected to primality tests, and if composite with $n\leq 11$, searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with $n\leq 7$ are now completely factored.

References [Enhancements On Off] (What's this?)

  • 1. H. Dubner and W. Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), 397-405. MR 95c:11010
  • 2. E. Lindelöf, Le Calcul des Résidus et ses Applications a la Théorie des Fonctions, Gauthier-Villars, Paris 1905, formula (3) on p. 78.
  • 3. H. Riesel, Some Factors of the Numbers $G_n=6^{2^n}+1$ and $H_n=10^{2^n}+1$, Math. Comp. 23 (1969), 413-415. MR 39:6813
  • 4. H. Riesel, Common Prime Factors of the Numbers $A_n=a^{2^n}+1$, BIT 9 (1969), 264-269. MR 41:3381
  • 5. H. Riesel and A. Björn, Generalized Fermat Numbers, in Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics, W. Gautschi, ed., Proc. Symp. Appl. Math. 48 (1994), 583-587, Amer. Math. Soc., Providence, R.I., 1994. MR 95j:11006
  • 6. H. Riesel, Summation of Double Series Using the Euler-MacLaurin Sum Formula, BIT 36 (1996), 860-862. CMP 97:04

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Additional Information

Anders Björn
Affiliation: Department of Mathematics, Linköping University, S-581 83 Linköping, Sweden

Hans Riesel
Affiliation: Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Keywords: Fermat numbers, primes, factorization
Received by editor(s): May 6, 1996
Received by editor(s) in revised form: September 19, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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