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ECM using Edwards curves
"... Abstract. This paper introduces GMPEECM, a fast implementation of the ellipticcurve method of factoring integers. GMPEECM is based on, but faster than, the wellknown GMPECM software. The main changes are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use twisted inverted E ..."
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Abstract. This paper introduces GMPEECM, a fast implementation of the ellipticcurve method of factoring integers. GMPEECM is based on, but faster than, the wellknown GMPECM software. The main changes are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use twisted inverted Edwards coordinates; (3) use signedslidingwindow addition chains; (4) batch primes to increase the window size; (5) choose curves with small parameters a, d, X1, Y1, Z1; (6) choose curves with larger torsion.
Sieving the positive integers by large primes
, 1988
"... Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if yt cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous ..."
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Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if yt cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous solution of the differential delay equation up&(u) =6p,(u 1), p&(u) = 1, 0 < I ( < 1. This generalizes work by Dickman, de Bruijn, and Hildebrand, who considered the case where Q consists of all primes (and 6 = 1).
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
A PAIR OF DIFFERENCE DIFFERENTIAL EQUATIONS OF EULERCAUCHY TYPE
"... Abstract. We study two classes of linear difference differential equations analogous to EulerCauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory an ..."
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Abstract. We study two classes of linear difference differential equations analogous to EulerCauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general EulerCauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity. 1.