Results 1  10
of
68
On the number of solutions of simultaneous Pell equations
"... It is proven that if a and b are distinct nonzero integers then the simultaneous Diophantine equations x 2 − az 2 =1, y 2 − bz 2 =1 possess at most three solutions in positive integers (x, y, z). Since there exist infinite families of pairs (a, b) for which the above equations have at least two solu ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
It is proven that if a and b are distinct nonzero integers then the simultaneous Diophantine equations x 2 − az 2 =1, y 2 − bz 2 =1 possess at most three solutions in positive integers (x, y, z). Since there exist infinite families of pairs (a, b) for which the above equations have at least two solutions, this result is not too far from the truth. If, further, u and v are nonzero integers with av − bu nonzero, then the more general equations x 2 − az 2 = u, y 2 − bz 2 = v are shown to have ≪ 2 min{ω(u),ω(v)} log (u  + v) solutions in integers, where ω(m) denotes the number of distinct prime factors of m and the implied constant is absolute. These results follow from a combination of techniques including simultaneous Padé approximation to binomial functions, the theory of linear forms in two logarithms and some gap principles, both new and familiar. Some connections to elliptic curves and related problems are briefly discussed.
Solving constrained Pell equations
 Math. Comp
, 1998
"... Abstract. Consider the system of Diophantine equations x 2 − ay 2 = b, P (x, y) =z 2,whereP is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an i ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
Abstract. Consider the system of Diophantine equations x 2 − ay 2 = b, P (x, y) =z 2,whereP is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases P (x, y) =cy 2 + d and P (x, y) =cx + d, which arise when looking for integer points on an elliptic curve with a rational 2torsion point. 1.
Simultaneous Pellian equations
, 1989
"... this paper we describe a method for finding integer solutions of simultaneous Pellian equations, that is, integer triples (x; y; z) satisfying equations of the form (x + f) ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
this paper we describe a method for finding integer solutions of simultaneous Pellian equations, that is, integer triples (x; y; z) satisfying equations of the form (x + f)
FPGA Implementation of Point Multiplication on Koblitz Curves using Kleinian Integers
 In Cryptographic Hardware and Embedded Systems, CHES 2006
, 2006
"... Abstract. We describe algorithms for point multiplication on Koblitz curves using multiplebase expansions of the form k = ∑ ±τ a (τ − 1) b and k = ∑ ±τ a (τ − 1) b (τ 2 − τ − 1) c. We prove that the number of terms in the second type is sublinear in the bit length of k, which leads to the first p ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Abstract. We describe algorithms for point multiplication on Koblitz curves using multiplebase expansions of the form k = ∑ ±τ a (τ − 1) b and k = ∑ ±τ a (τ − 1) b (τ 2 − τ − 1) c. We prove that the number of terms in the second type is sublinear in the bit length of k, which leads to the first provably sublinear point multiplication algorithm on Koblitz curves. For the first type, we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of τadic nonadjacent form expansions. We present details of an innovative FPGA implementation of our algorithm and performance data demonstrating the efficiency of our method. 1
On two classes of simultaneous Pell equations with no solutions
 Math. Comp
, 1999
"... Abstract. In this paper we describe two classes of simultaneous Pell equations of the form x 2 − dy 2 = z 2 − ey 2 = 1 with no solutions in positive integers x, y, z. The proof is elementary and covers the case (d, e) =(8,5), which was solved by E. Brown using very deep methods. 1. ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Abstract. In this paper we describe two classes of simultaneous Pell equations of the form x 2 − dy 2 = z 2 − ey 2 = 1 with no solutions in positive integers x, y, z. The proof is elementary and covers the case (d, e) =(8,5), which was solved by E. Brown using very deep methods. 1.
CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
, 2003
"... The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and GrossZagier, the task was a finite decision problem for any N. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and GrossZagier, the task was a finite decision problem for any N. Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work for other small class numbers, up to 100, say.” However, more than ten years later, after doing N =5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N ≤ 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the GoldfeldOesterlé work, which used an elliptic curve Lfunction with an order 3 zero at the central critical point, to instead consider Dirichlet Lfunctions with lowheight zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of MontgomeryWeinberger. Our method is still quite computerintensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large “exceptional modulus” of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.
N.J.A.Sloane, Complex and Integral Laminated Lattices
 Transactions of the American Mathematical Society
"... transactions of the american mathematical society ..."
Diophantine approximations, Diophantine equations, Transcendence and Applications
 Indian J. Pure Appl. Math
"... This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c> 0 depend ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c> 0 depending only on α such that  α − p c q qn for all rational numbers p q with q> 0. This inequality is from diophantine approximations. Any nontrivial improvement of this inequality shows that certain class of diophantine equations, known as Thue equations, has only finitely many integral solutions. Also, the above inequality can be applied to establish the transcendence of ∞ ∑ 1 numbers like. For an other example on connection between these topics, we 2n! n=1 refer to an account on equation (2) in this article.