Three new factors of Fermat numbers

Brent, Richard P.; Dilcher, Karl; Halewyn, C. van

doi:10.1090/s0025-5718-00-01207-2

Cited by 7 publications

(11 citation statements)

References 27 publications

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“…Fermat Numbers 73 F 8 : factor 1 Brent and Pollard (1980) F 9 : factor 1 Western (1903), other factors A.K. In this regard, the articles of Brent (1999), and of Brent, Crandall, Dilcher & van Halewyn (2000) are very informative. In this regard, the articles of Brent (1999), and of Brent, Crandall, Dilcher & van Halewyn (2000) are very informative.…”

Section: Proof If (I) Is Assumed Then By Euler's Criterion For the mentioning

confidence: 99%

The Little Book of Bigger Primes

Ribenboim¹

2004

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Section: Proof If (I) Is Assumed Then By Euler's Criterion For the mentioning

confidence: 99%

The Little Book of Bigger Primes

Ribenboim¹

2004

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“…2. 19 Let α(x, y) = (p(x)/q(x), y •s(x)/t(x)) be an endomorphism of the elliptic curve E given by y 2 = x 3 + Ax + B, where p, q, s, t are polynomials such that p and q have no common root and s and t have no common root.…”

Section: 7mentioning

confidence: 99%

Elliptic curves: number theory and cryptography

2004

Choice Reviews Online

101

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present book, but contains few proofs. It should be consulted by serious students of elliptic curve cryptography. We hope that the present book provides a good introduction to and explanation of the mathematics used in that book. The books by Enge [38], Koblitz [66], [65], and Menezes [82] also treat elliptic curves from a cryptographic viewpoint and can be profitably consulted.Notation. The symbols Z, F q , Q, R, C denote the integers, the finite field with q elements, the rationals, the reals, and the complex numbers, respectively. We have used Z n (rather than Z/nZ) to denote the integers mod n. However, when p is a prime and we are working with Z p as a field, rather than as a group or ring, we use F p in order to remain consistent with the notation F q . Note that Z p does not denote the p-adic integers. This choice was made for typographic reasons since the integers mod p are used frequently, while a symbol for the p-adic integers is used only in a few examples in Chapter 13 (where we use O p ). The p-adic rationals are denoted by Q p . If K is a field, then K denotes an algebraic closure of K. If R is a ring, then R × denotes the invertible elements of R. When K is a field, K × is therefore the multiplicative group of nonzero elements of K. Throughout the book, the letters K and E are generally used to denote a field and an elliptic curve (except in Chapter 9, where K is used a few times for an elliptic integral). Acknowledgments. The author thanks Bob Stern of CRC Press forsuggesting that this book be written and for his encouragement, and the editorial staff at CRC Press for their help during the preparation of the book.

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“…Because we can perform the reductions mod 2 n ± 1 using binary shift and add/subtract operations, which are much faster (for large n) than multiply or divide operations, a significant speedup may be possible. This idea was not implemented in programs B-D, but was used successfully in programs which found factors of F 13 and F 16 , see [13,17].…”

Section: The Multiplication Algorithmmentioning

confidence: 99%

“…Because the F n grow rapidly in size, a method which factors F n may be inadequate for F n+1 . Historical details and references can be found in [21,35,36,44,74], and some recent results are given in [17,26,27,34].…”

Section: Introduction and Historical Summarymentioning

confidence: 99%

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Factorization of the tenth Fermat number

Brent

1999

Math. Comp.

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Abstract. 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 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , . . . , F 11 .

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Three new factors of Fermat numbers

Cited by 7 publications

References 27 publications

The Little Book of Bigger Primes

The Little Book of Bigger Primes

Elliptic curves: number theory and cryptography

Factorization of the tenth Fermat number

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