Abstract
Great strides have been made in recent years in the search for ever larger prime Generalized Fermat Numbers (GFN). We briefly review the history of the GFN prime search, and describe new implementations of the ‘Genefer’ software (now available as open source) using CUDA and optimised CPU assembler which have underpinned this unprecedented progress. The results of the ongoing search are used to extend Gallot and Dubner’s published tables comparing the theoretical predictions with actual distributions of primes, and we report on recent discoveries of GFN primes with over one million digits.
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Acknowledgments
The first author acknowledges the support of NAIS, the Centre for Numerical Algorithms and Intelligent Software (EPSRC grant EP/G036136/1).
We also wish to thank several people who have contributed to the GFN prime search. First, we thank Yves Gallot for popularising the search, developing the initial Genefer code upon which the entire project is based, and also for useful discussions concerning the purported excess of primes at large \(n\) (see Sect. 3). Second, we thank David Underbakke, Mark Rodenkirch, Ken Brazier, Shoichiro Yamada, Ronald Schneider and Anand Nair, who have all contributed to the ongoing development of the PRP testing and sieving software. Third, thanks go to the PrimeGrid team Rytis Slatkevicius, Lennart Vogel and John Blazek without whom the search would not have reached such a wide audience. Finally, we are grateful to all the ‘crunchers’ who have dedicated their computer resources and made possible the ongoing success of the search.
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Bethune, I., Goetz, M. (2014). Extending the Generalized Fermat Prime Number Search Beyond One Million Digits Using GPUs. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2013. Lecture Notes in Computer Science(), vol 8384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55224-3_11
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DOI: https://doi.org/10.1007/978-3-642-55224-3_11
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