Distribution of generalized Fermat prime numbers

Abstract

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.

Mathematics of Computation

“…The evaluation of a (N −1)/2 using the left-to-right binary algorithm takes only log 2 n modular squaring operations [9, p. 9], so the key to doing this quickly is in multiplying quickly. Proth.exe multiplies by evaluating the convolution of the polynomials defined by the representation of the numbers in base 2 16 , and the convolutions are evaluated using real-signal Fast Fourier Transforms implemented with double precision floating-point numbers [34, chapter 20]. In modern microprocessors, accessing the data in the main memory is more than 20 times slower than executing an arithmetic operation, so algorithms such as split-radix FFT, which were developed to minimize the number of multiplications, are no longer the best algorithms.…”

“…For example, as of the date we wrote this article, this program was used to find the following record prime numbers: (1) 169719 · 2 557557 + 1 (167847 digits), the largest known "non-Mersenne" prime (found by Stephen Scott in 2000); (2) 481899 · 2 481899 + 1, the largest known Cullen prime [23] (found by Masakatu Morii in 1998); (3) 151023 · 2 151023 − 1, the largest known Woodall prime [23] (found by Kevin O'Hare in 1998); (4) 506664 16384 + 1, the largest known Generalized Fermat prime [4,16] 3.2. Primality Proving.…”

Genefer: Programs for Finding Large Probable Generalized Fermat Primes

“…The code implements a Fermat primality test, computing a p-1 (mod p) for a candidate GFN p. It is a necessary, but not sufficient, condition for p prime that this test returns 1, so Genefer pseudo-primes must be subsequently tested for primality using a different program such as PFGW [1]. Genefer was originally developed by the second author, who managed a manual distributed computing effort between 2000 and 2004, which discovered several 100,000 + digit primes [2]. Since 2009, the PrimeGrid [3] volunteer computing project has used Genefer to extend the search to well over a million digits [4] and is currently searching for a world-record sized GFN prime.…”

Die Welt der Primzahlen

“…Aber wie lange wird dieser Frieden andauern?WieDubner & Gallot (2002) in ihrem Artikel erläutern, gibt es erwartungsgemäß sehr viel mehr verallgemeinerte Fermat-Primzahlen vergleichbarer Größenordnung, und daher könnte nach ihren Aussagen eine gut organisierte Suche die Rangordnung unter den größten bekannten Primzahlen schon bald verändern. Noll, B. Parady, G. Smith, J. Smith und S. Zarantonello, die Mersenne-Primzahl M 216091 entthront hatte.…”