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

DOI:
https://doi.org/10.1090/S0025-5718-98-00891-6

Erratum:
Math. Comp. 74 (2005), 2099

Erratum:
Math. Comp. 80 (2011), 1865-1866

Supplement:
Additional information related to this article.

MathSciNet review:
1433262

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A search for prime factors of the generalized Fermat numbers has been carried out for all pairs with and GCD. The search limit on the factors, which all have the form , was for and for . Many larger primes of this form have also been tried as factors of . Several thousand new factors were found, which are given in our tables.-For the smaller of the numbers, i.e. for , or, if , for , the cofactors, after removal of the factors found, were subjected to primality tests, and if composite with , searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with are now completely factored.

**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 and ,*Math. Comp.**23**(1969), 413-415. MR**39:6813****4.**H. Riesel,*Common Prime Factors of the Numbers ,*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**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
11-04,
11A51,
11Y05,
11Y11

Retrieve articles in all journals with MSC (1991): 11-04, 11A51, 11Y05, 11Y11

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

DOI:
https://doi.org/10.1090/S0025-5718-98-00891-6

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