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Factorization Of The Tenth Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
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Cited by 22 (10 self)
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We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) ..."
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Cited by 14 (5 self)
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For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) = C 2 n nxD 2 n ; Here A n (x); : : : ; D n (x) are polynomials with integer coecients. We show how these coef cients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n (x); : : : ; D n (x), and illustrate the application to integer factorization with some numerical examples.
Computing Aurifeuillian factors
 In Computational Algebra and Number Theory, Mathematics and its Applications Vol. 325
, 1995
"... Abstract. For odd squarefree n> 1, the cyclotomic polynomial Φn(x) satisfies an identity Φn(x) = Cn(x) 2 ± nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas. Here Cn(x) and Dn(x) are monic polynomials with integer coefficients. These coefficients can be computed by simple algorithms which require O(n ..."
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Cited by 1 (0 self)
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Abstract. For odd squarefree n> 1, the cyclotomic polynomial Φn(x) satisfies an identity Φn(x) = Cn(x) 2 ± nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas. Here Cn(x) and Dn(x) are monic polynomials with integer coefficients. These coefficients can be computed by simple algorithms which require O(n 2) arithmetic operations over the integers. Also, there are explicit formulas and generating functions for Cn(x) and Dn(x). This paper is a preliminary report which states the results for the case n = 1 mod 4, and gives some numerical examples. The proofs, generalisations to other squarefree n, and similar results for the identities of Gauss and Dirichlet, will appear elsewhere. 1.
Factorizations of a^n ± 1, 13 ≤ a < 100: Update 2
, 1996
"... This Report updates the tables of factorizations of a n \Sigma 1 for 13 a ! 100, previously published as CWI Report NMR9212 (June 1992) and updated in CWI Report NMR9419 (September 1994). A total of 760 new entries in the tables are given here. The factorizations are now complete for n ! 67, an ..."
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This Report updates the tables of factorizations of a n \Sigma 1 for 13 a ! 100, previously published as CWI Report NMR9212 (June 1992) and updated in CWI Report NMR9419 (September 1994). A total of 760 new entries in the tables are given here. The factorizations are now complete for n ! 67, and there are no composite cofactors smaller than 10 94 . 1991 Mathematics Subject Classification. Primary 11A25; Secondary 1104 Key words and phrases. Factor tables, ECM, MPQS, SNFS To appear as Report NMR96??, Centrum voor Wiskunde en Informatica, Amsterdam, March 1996. Copyright c fl 1996, the authors. Only the front matter is given here. For the tables, see rpb134u2.txt . rpb134u2 typeset using L a T E X 1 Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with a n ...
On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree n ? 1 the cyclotomic polynomial \Phi n (x) satisfies the identity of Gauss 4\Phi n (x) = A 2 n \Gamma (\Gamma1) (n\Gamma1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is \Phi n ((\Gamma1) (n\Gamma1)=2 x) = C 2 n \Gamma nxD 2 n or, in the case ..."
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For odd squarefree n ? 1 the cyclotomic polynomial \Phi n (x) satisfies the identity of Gauss 4\Phi n (x) = A 2 n \Gamma (\Gamma1) (n\Gamma1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is \Phi n ((\Gamma1) (n\Gamma1)=2 x) = C 2 n \Gamma nxD 2 n or, in the case that n is even and squarefree, \Sigma\Phi n=2 (\Gammax 2 ) = C 2 n \Gamma nxD 2 n ; Here An (x); : : : ; Dn (x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for An (x); : : : ; Dn (x), and illustrate the application to integer factorization with some numerical examples.