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

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.**Harvey Dubner and Wilfrid Keller,*Factors of generalized Fermat numbers*, Math. Comp.**64**(1995), no. 209, 397–405. MR**1270618**, 10.1090/S0025-5718-1995-1270618-1**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.**Hans Riesel,*Some factors of the numbers 𝐺_{𝑛}=6²ⁿ+1 and 𝐻_{𝑛}=10²ⁿ+1*, Math. Comp.**23**(1969), 413–415. MR**0245507**, 10.1090/S0025-5718-1969-0245507-6**4.**Hans Riesel,*Common prime factors of the numbers 𝐴_{𝑛}=𝑎^{2ⁿ}+1*, Nordisk Tidskr. Informationsbehandling (BIT)**9**(1969), 264–269. MR**0258735****5.**Hans Riesel and Anders Björn,*Generalized Fermat numbers*, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 583–587. MR**1314895**, 10.1090/S0025-5718-10-02371-9**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