Distribution of generalized Fermat prime numbers

Dubner, Harvey; Gallot, Yves

doi:10.1090/S0025-5718-01-01350-3

Distribution of generalized Fermat prime numbers
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by Harvey Dubner and Yves Gallot PDF

Math. Comp. 71 (2002), 825-832 Request permission

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 Sierpiński. A list of the current largest known GFN primes is included.

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