Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.

This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a non-trivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is

In this work we investigate the di culty of the discrete logarithm problem in class groups of imaginary quadratic orders. In particular, we discuss several strategies to compute discrete logarithms in those class groups. Based on heuristic reasoning, we give advice for selecting the cryptographic parameter, i.e. the discriminant, such that cryptosystems based on class groups of imaginary quadratic orders would o er a similar security as commonly used cryptosystems. 1

In this paper, we apply Pad'e approximation methods to derive completely explicit measures of irrationality for certain classes of algebraic numbers. Our approach is similar to that taken previously by G.V. Chudnovsky but has some fundamental advantages with regards to determining implicit constants. Our general results may be applied to produce specific bounds of the flavour of fi fi fi fi 3 p 2 \Gamma p q fi fi fi fi ? 1 4 q \Gamma2:45 and fi fi fi fi 7 p 5 \Gamma p q fi fi fi fi ? 1 4 q \Gamma4:43 which we show to hold for any nonzero integers p and q. Further examples are tabulated and applications to Diophantine equations are briefly discussed as are other topics of related interest. 1

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Presidential Faculty Fellow. His research is partiallly supported by the NSF. The research of the second two authors is partially supported by NSERC of Canada 1 1 Introduction Let D be the fundamental discriminant of a real quadratic field and let S = f5; 8; 12; 13; 17; 24; 28; 33; 40; 57; 60; 73; 76; 88; 97; 105; 124; 129; 136; 145; 156; 184; 204; 249; 280; 316; 345; 364; 385; 424; 456; 520; 609; 616; 924; 940; 984; 1065; 1596; 2044; 2244; 3705g: At the end of Chapter 6 of [5], the second author made the following conjecture. Conjecture. The values of D for which the least prime p such that the Kronecker symbol (D=p) = \Gamma1 satisfies p ? p D=2 are precisely those in S. He also veri...

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The author uses an elementary lemma on primes dividing binomial coefficients and estimates for primes in arithmetic progressions to sharpen a theorem of J. Rickert on simultaneous approximation to pairs of algebraic numbers. In particular, it is proven that max aefi fi fi fi p 2 \Gamma p 1 q fi fi fi fi ; fi fi fi fi p 3 \Gamma p 2 q fi fi fi fi oe ? 10 \Gamma10 q \Gamma1:8161 for p 1 ; p 2 and q integral. Applications of these estimates are briefly discussed. 1 1

Let OK be any domain with field of fractions K. LetF(x, y) ∈ OK[x, y] be a homogeneous polynomial of degree n, coprime to y, andassumedto have unit content (i.e., the coefficients of F generate the unit ideal in OK). Assume that gcd(n, char(K)) = 1. Let h ∈ OK and assume that the polynomial hz n − F(x, y) is irreducible in K[x, y, z]. We denote by X F,h/K the nonsingular complete model of the projective plane curve CF,h/K defined by the equation hz n − F(x, y) = 0. We shall assume in this article that g(X F,h) ≥ 2. When K is a number field, Mordell’s Conjecture (now Faltings ’ Theorem) implies that |X F,h(K) | < ∞. Caporaso, Harris, and Mazur ([CHM, 1.1]) have shown that if Lang’s conjecture for varieties of general type is true, then for any number field K, thesize|X(K) | of the set of K-rational points of any curve X/K of genus g(X) ≥ 2 can be bounded by a constant depending only on g(X). Prior to the paper [CHM], Mazur and others had asked whether |X(K) | can be bounded by a constant depending only on

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The author uses irrationality and linear independence measures for certain algebraic numbers to derive explicit upper bounds for the solutions of related norm form equations. The Lenstra-Lenstra-Lov'asz lattice basis reduction algorithm is then utilized to show that the integer solutions to NK=Q (x 4 p N 4 \Gamma 1 + y 4 p N 4 + 1 + z) = \Sigma1 (where K = Q( 4 p N 4 \Gamma 1; 4 p N 4 + 1)) are given by (x; y; z) = (0; 0; \Sigma1), (\Sigma1; 0; \SigmaN ) and (0; \Sigma1; \SigmaN ), for 5 N 100. 1