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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 ..."
Abstract

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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Summary
"... Human immunoglobulin M (IgM) rheumatoid factors (RFs) show primary direct enzymelinked immunosorbent assay (ELISA) reactivity with Fab rather than Fc fragments of monoclonal antibody (mAb) II481 directed against the Fc3'binding site of herpes simplex virus glycoprotein gE. This preferential anti ..."
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Human immunoglobulin M (IgM) rheumatoid factors (RFs) show primary direct enzymelinked immunosorbent assay (ELISA) reactivity with Fab rather than Fc fragments of monoclonal antibody (mAb) II481 directed against the Fc3'binding site of herpes simplex virus glycoprotein gE. This preferential antiFab specificity suggests that RFs react with antigenbinding portions of mAb II481 as antiidiotypic antibodies directed at the combining site regions of mAb reacting with the Fc3'binding region of gE. Analysis of this idiotypeantiidiotype reaction employed polymerase chain reaction amplification and sequencing of the variable heavy and light (VH and VL) regions of mAb 11481. When VH and VL regions of mAb II481 were synthesized as overlapping 7met peptides on polypropylene pins, a panel of 10 polyclonal and 6 rnonoclonal human IgM RFs reacted primarily with epitopes within the three solventexposed mAb 11481 complementarity determining regions (CDRs). Preincubation of single CDR heptamer peptides with IgM RFs in free solution, resulted in 63100 % inhibition of RF binding to mAb II481 on the ELISA plate, confirming the antigenic importance of linear CDR regions for RF reactivity. Combinations of two or three CDR peptides frequently produced 94100 % inhibition of RF binding to whole mAb II481. Control peptides, singly or in combination, showed no inhibition. Computer modeling
unknown title
, 2004
"... Foxl2 disruption causes mouse ovarian failure by pervasive blockage of follicle development ..."
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Foxl2 disruption causes mouse ovarian failure by pervasive blockage of follicle development
On the Infinitude of Some Special Kinds of Primes
, 2009
"... The aim of this paper is to try to establish a generic model for the problem that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–f ..."
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The aim of this paper is to try to establish a generic model for the problem that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–from Euclid’s second theorem to GreenTao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, generalized Euler’s function, the primecounting function and SchinzelSierpinski’s Conjecture and so on, obtained an analogy of the Chinese Remainder Theorem. By proposed obtrusively several conjectures, we gave a new way for determining the existence of some special kinds of primes. Finally, we proposed sufficient and necessary conditions that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it.