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Distribution of generalized Fermat prime numbers

Author(s): Harvey Dubner; Yves Gallot.
Journal: Math. Comp. 71 (2002), 825-832.
MSC (2000): Primary 11Y11; Secondary 11A41
Posted: May 17, 2001
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Abstract | References | Similar articles | Additional information

Abstract:

Numbers of the form $F_{b,n}=b^{2^n}+1$ are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form $2^m-1$. The theoretical distributions of GFN primes, for fixed $n$, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn's quantitative form of ``Hypothesis H" of Schinzel and Sierpinski. A list of the current largest known GFN primes is included.

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Additional Information:

Harvey Dubner
Affiliation: 449 Beverly Road, Ridgewood, New Jersey 07450
Email: hdubner1@compuserve.com

Yves Gallot
Affiliation: 12 bis rue Perrey, 31400 Toulouse, France
Email: galloty@wanadoo.fr

DOI: 10.1090/S0025-5718-01-01350-3
PII: S 0025-5718(01)01350-3
Keywords: Prime numbers, generalized Fermat numbers, primality proving algorithms
Received by editor(s): October 13, 1999
Received by editor(s) in revised form: July 10, 2000
Posted: May 17, 2001
Copyright of article: Copyright 2001, American Mathematical Society

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