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Mathematics of Computation

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Distribution of generalized Fermat prime numbers

Authors: Harvey Dubner and Yves Gallot
Journal: Math. Comp. 71 (2002), 825-832
MSC (2000): Primary 11Y11; Secondary 11A41
Published electronically: May 17, 2001
MathSciNet review: 1885631
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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

Yves Gallot
Affiliation: 12 bis rue Perrey, 31400 Toulouse, France

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
Published electronically: May 17, 2001
Article copyright: © Copyright 2001 American Mathematical Society