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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 ..."
Abstract

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 .
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...
Two new factors of Fermat numbers
, 1997
"... Abstract. We report the discovery of new 27decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19 ..."
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Cited by 5 (2 self)
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Abstract. We report the discovery of new 27decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19694 decimal digits respectively. 1.
Three New Factors of Fermat Numbers
 Math. Comp
, 2000
"... We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are ..."
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Cited by 4 (0 self)
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We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively. 1.
Let
, 1986
"... It is well known (see, for example, Ex. 3.96 of [1]) that the polynomials x 2 ' 3 ° + x 3J + 1 are irreducible in GF(2)[x] for J = 0, 1, 2,.... Since (x 2 ' 3 ' + x 3J + l)(x 3J + 1) = x 3J+ ± + 1 is a squarefree polynomial, it follows that the period of each root of a; 2 * 3 + x + ..."
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It is well known (see, for example, Ex. 3.96 of [1]) that the polynomials x 2 ' 3 ° + x 3J + 1 are irreducible in GF(2)[x] for J = 0, 1, 2,.... Since (x 2 ' 3 ' + x 3J + l)(x 3J + 1) = x 3J+ ± + 1 is a squarefree polynomial, it follows that the period of each root of a; 2 * 3 + x + 1 is precisely 3 J 1, only one and a half times the degree of the polynomial. The field Cj ~ GF(2)[x]/(x 2 ' 3J ' + x 3 ' + 1) ~ GF(2 2 ' 3J) may be obtained by iterated cubic extensions beginning with CQ ~ GF(2)(xQ)9 where x0 £ 1 is a cube root of unity. We have C1 ~ CQCXL), where x x is any solution to x1 = xQ, Iterating, C+1 « CAxj + 1) 9 where x + 1 = x. This paper deals with an iterated quadratic extension of GF(2), whose generators are described by