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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Factors of generalized Fermat numbers
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by Anders Björn and Hans Riesel PDF
Math. Comp. 67 (1998), 441-446 Request permission

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.
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Additional Information
  • Anders Björn
  • Affiliation: Department of Mathematics, Linköping University, S-581 83 Linköping, Sweden
  • Email: anbjo@mai.liu.se
  • Hans Riesel
  • Affiliation: Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden
  • Email: riesel@nada.kth.se
  • Received by editor(s): May 6, 1996
  • Received by editor(s) in revised form: September 19, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 441-446
  • MSC (1991): Primary 11-04, 11A51, 11Y05, 11Y11
  • DOI: https://doi.org/10.1090/S0025-5718-98-00891-6
  • MathSciNet review: 1433262