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Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
Integer Factorization Algorithms Illustrated by the Factorization of Fermat Numbers
, 1998
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A History of Factor Tables with Notes on the Birth of Number Theory 1657–1817
"... The history of the construction, organisation and publication of factor tables from 1660 to 1817, in itself a fascinating story, also touches upon many topics of general interest for the history of mathematics. The considerable labour involved in constructing and correcting these tables has pushed m ..."
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The history of the construction, organisation and publication of factor tables from 1660 to 1817, in itself a fascinating story, also touches upon many topics of general interest for the history of mathematics. The considerable labour involved in constructing and correcting these tables has pushed mathematicians and calculators to organise themselves in some network. Around 1660 J. Pell was the first to motivate others to calculate a large factor table, for which he saw many applications, not only in Diophantine analysis but also in arithmetic and even philosophy. Some hundred years later (1770), J.H. Lambert launched a table project that would engage many computers and mathematicians to (re)produce Pell’s table and extend it. Importantly, Lambert also pointed out that a theory of numbers, of divisors and factoring methods was still lacking. Lambert’s ideas were taken up by his colleagues at the Berlin Academy, and indirectly by L. Euler in St Petersburg. Finally, the many numbertheoretical essays that were written in the context of Lambert’s table project contributed importantly to the birth of higher arithmetic around 1800, starting with A.M. Legendre’s and C.F. Gauss’s work. Une histoire des Tables des Diviseurs, avec des Notes sur la Naissance de la Théorie des Nombres 1657–1817 L’histoire de la fabrication, l’organisation et la publication des tables de diviseurs, de 1600 à 1817, offre en et pour soi une histoire
The concrete theory of numbers: Problem of simplicity of Fermat numbertwins
, 2008
"... The problem of simplicity of Fermat numbertwins f ± n = 2 2n ± 3 is studied. The question for what n numbers f ± n are composite is investigated. The factoridentities for numbers of a kind x 2 ± k are found. 1 ..."
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The problem of simplicity of Fermat numbertwins f ± n = 2 2n ± 3 is studied. The question for what n numbers f ± n are composite is investigated. The factoridentities for numbers of a kind x 2 ± k are found. 1