Attila Pethö

Perfect powers in second order linear recurrences

Abstract Let A, B, G0, G1 be integers, and Gn = AGn − 1 − BGn − 2 for n ≥ 2. Let further S be the set of all nonzero integers composed of primes from some fixed finite set. In this paper we shall prove that natural conditions for A, B, G0 and G1 imply, that the diophantine equation Gn = wxq has only finitely many solutions in integers ∥x∥ > 1, q ≥ 2, n and w ∈ S.

Simple families of Thue inequalities

We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.

On Mordell's equation

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordells Equation y2=x3+k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordells Equation in Z for all integers k within the range 0 < | k | ≤ 10 000 and partially extend the computations to 0 < | k | ≤ 100 000. For these values of k, the constant in Halls conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

On Digit Expansions with Respect to Linear Recurrences

Abstract An explicit formula for the mean value of the sum-of-digits function with respect to linear recurring sequences is established. Thus a recent paper of J. Coquet and P. Van Den Bosch on the Fibonacci number system is extended to the general case.

On the resolution of index form equations in biquadratic number fields, II

In this paper we develop a method for computing all small solutions (i.e. with coordinates of absolute value <107) of index form equations in totally real biquadratic number fields. If the index form equation is not solvable, this will also be recognized by our algorithm in most cases. As an application we present all such solutions in quadratic extensions K of Q(√5) of discriminant DKQ < 63000 and of Q(√2) of discriminant DKQ < 39000.

Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation f a (x,y) = x 4 − ax 3 y − x 2 y 2 + axy 3 + y 4 = 1 for |a| > 3 has exactly 12 solutions except when |a| = 4, when it has 16 solutions. If α = α(a) denotes one of the zeros of f a (x,1), then for |a| ≥ 4 we also find all γ ∈ Z[a] with Z[γ] = Z[α].

Polynomial values in linear recurrences, II☆

Abstract We prove a necessary condition for the Diophantine equation G m = P ( x ), with G m a second order linear recurrence sequence and P(x) ∈ Z [ x ], to have infinitely many integral solutions, x , m .

Computing all S -integral points on elliptic curves

In this note we combine the advantages of the methods of Siegel-Baker-Coates and of Lang-Zagier for the computation of S-integral points on elliptic curves in Weierstrass normal form over the rationals. In this way we are able to overcome the absence of an explicit lower bound for linear forms in q-adic elliptic logarithms. We present an efficient algorithm for determining all S-integral points on such curves.

Zur Transzendenz gewisser Reihen

From Schmidts simultaneous approximation theorem we deduce transcendence results concerning series of rational numbers. The denominators of these numbers are from finitely many linear recursive sequences and have to satisfy a divisibility as well as a growth condition. (In an appendix the second author studies the connections between these two kinds of hypothesis.) For the numerators we need some growth conditions too. We study also the implications of Mahlers analytic transcendence method from 1929 to the arithmetical questions considered mainly.

Generalized radix representations and dynamical systems III

For r = (r1, . . . , rd) ∈ Rd the map τr : Zd → Zd given by τr(a1, . . . , ad) = (a2, . . . , ad,−br1a1 + · · ·+ rdadc) is called a shift radix system if for each a ∈ Zd there exists an integer k > 0 with τk r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like β-expansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials