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30
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 ..."
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Cited by 33 (6 self)
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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 ..."
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Cited by 23 (0 self)
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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.
On some computational problems in finite abelian groups
 Mathematics of Computation
, 1997
"... Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the Oconstants and Ωconstants. We implemented the algorithms for class ..."
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Cited by 23 (7 self)
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Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the Oconstants and Ωconstants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results. Our algorithms are based on a modification of Shanks ’ babystep giantstep strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order. 1.
SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES
, 2008
"... Abstract. We study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients (by a result of Nikulin this action is independent on the choice ..."
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Cited by 13 (8 self)
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Abstract. We study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients (by a result of Nikulin this action is independent on the choice of the K3 surface). With the help of elliptic fibrations we determine the invariant sublattice and its perpendicular complement, and show that the latter coincides with the CoxeterTodd lattice in the case of automorphism of order three. In the paper [Ni1] Nikulin studies finite abelian groups G acting symplectically (i.e. G H2,0(X,C) = id H2,0(X,C)) on K3 surfaces (defined over C). One of his main result is that the action induced by G on the cohomology group H2 (X, Z) is unique up to isometry. In [Ni1] all abelian finite groups of automorphisms of a K3 surface acting symplectically
Archimedes' Cattle Problem
 American Mathematical Monthly
, 1998
"... this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of ..."
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Cited by 9 (3 self)
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this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of the problem is
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
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. ..."
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Cited by 8 (0 self)
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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.
Topological Conjugacy of Linear Endomorphisms of the 2Torus
 Trans. Amer. Math. Soc
, 1997
"... . We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the twodimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two can ..."
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Cited by 6 (1 self)
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. We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the twodimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the LatimerMacDuffeeTaussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classifi...
Nonsymplectic automorphisms of order 3 on K3 surfaces
, 2008
"... Abstract. In this paper we study K3 surfaces with a nonsymplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli sp ..."
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Cited by 4 (0 self)
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Abstract. In this paper we study K3 surfaces with a nonsymplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.
SQUARE FORM FACTORIZATION
, 2007
"... We present a detailed analysis of SQUFOF, Daniel Shanks’ Square Form Factorization algorithm. We give the average time and space requirements for SQUFOF. We analyze the effect of multipliers, either used for a single factorization or when racing the algorithm in parallel. ..."
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Cited by 4 (0 self)
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We present a detailed analysis of SQUFOF, Daniel Shanks’ Square Form Factorization algorithm. We give the average time and space requirements for SQUFOF. We analyze the effect of multipliers, either used for a single factorization or when racing the algorithm in parallel.