Abstract not found

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.

] C.J. Smyth. On measures of polynomials in several variables. Bull. Australian Math. Soc., 23:49--63, 1981. Corrigendum: G. Myerson and C.J. Smyth, 26 (1982), 317-319. [soule-1991] C. Soul'e. Geometrie d'Arakelov et th'eorie des nombres transcendants. Ast'erisque, 198-200:355--371, 1991. [stewart-1977] C.L. Stewart. On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. Proceedings of the London Math. Soc., 35:425--447, 1977. [stewart-1977-78] C.L. Stewart. On a theorem of Kronecker and a related question of Lehmer. In S'eminaire de Th'eorie de Nombres Bordeaux 1977/78. Birkhauser, Basel, 1978. [stewart-1978] C.L. Stewart. Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France, 106:169--176, 1978. [szydlo-1985] B. Szydlo. An application of some theorems of G. Szegoe to Mahler measure of polynomials. Discuss. Math., 7:145--148, 1985. [tate-thesis]

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 2-torsion point. 1.

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. 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.

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 Gross-Zagier, 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 Goldfeld-Oesterlé work, which used an elliptic curve L-function with an order 3 zero at the central critical point, to instead consider Dirichlet L-functions with low-height 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 Montgomery-Weinberger. Our method is still quite computer-intensive, 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.

Abstract not found

An m-cycle of the 3n+1-problem is defined as a periodic orbit with m local minima. In this article we derive lower and upper bounds for the cycle length and the elements of (hypothetical) m-cycles.

In this thesis we consider sequences of non-negative integers which arise from counting the periodic points of a map T: X → X, where X is a non-empty set. Some of the main results obtained are concerned with the counting of the periodic points of an endomorphism of a group, in particular when the group is locally nilpotent, for which class of groups a local-global property is established. The ideas developed are applied to some classical sequences, including the Bernoulli and Euler numbers, which are shown to have certain ‘dynamical’ properties. We also consider the Lehmer-Pierce construction for sequences of integers, looking at possible generalizations and their associated measures.