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

DOI:
https://doi.org/10.1090/S0025-5718-01-01350-3

Published electronically:
May 17, 2001

MathSciNet review:
1885631

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Numbers of the form 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 . The theoretical distributions of GFN primes, for fixed , 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:
https://doi.org/10.1090/S0025-5718-01-01350-3

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

Published electronically:
May 17, 2001

Article copyright:
© Copyright 2001
American Mathematical Society