Kálmán Győry

Bounds for the solutions of S-unit equations and decomposable form equations

In this paper we improve upon in terms of S the best known effective upper bounds for the solutions of S-unit equations and decomposable form equations.

Perfect powers from products of consecutive terms in arithmetic progression

We prove that for any positive integers x, d, k with gcd(x, d) = 1 and 3 11, a great number of new ternary equations arise that we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the different cases in the usual manner. The main novelty of this paper is that we algorithmize our proofs which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the arising new ternary equations with several sieves based on the ternary equations already solved. In this way we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger k as well, but there is of course a computational time limit.

Effective results for unit equations over finitely generated integral domains

Let A ⊃ Z be an integral domain which is finitely generated over Z and let a,b,c be non-zero elements of A. Extending earlier work of Siegel, Mahler and Parry, in 1960 Lang proved that the equation (*) ae + bη = c in e, η ∈ A ∗ has only finitely many solutions. Using Baker’s theory of logarithmic forms, Gyý ory proved, in 1979, that the solutions of (*) can be determined effectively if A is contained in an algebraic number field. In this paper we prove, in a quantitative form, an effective finiteness result for equations (*) over an arbitrary integral domain A of characteristic 0 which is finitely generated over Z. Our main tools are already existing effective finiteness results for (*) over number fields and function fields, an effective specialization argument developed by Gyý in the 1980’s, effective results of Hermann (1926) and Seidenberg (1974) on linear equations over polynomial rings over fields, and similar such results by Aschenbrenner, from 2004, on linear equations over polynomial rings over Z. We prove also an effective result for the exponential equation

The Divisibility of an – bn by Powers of n

Abstract For given integers a, b and j ≥ 1 we determine the set of integers n for which an – bn is divisible by nj . For j = 1, 2, this set is usually infinite; we determine explicitly the exceptional cases for which a, b the set is finite. For j = 2, we use Zsigmondys Theorem for this. For j ≥ 3 and gcd(a, b) = 1, is probably always finite; this seems difficult to prove, however. We also show that determination of the set of integers n for which an an + bn is divisible by nj can be reduced to that of .

Representation of finite graphs as difference graphs of S-units, I

Abstract Let S be a finite non-empty set of primes, Z S the ring of those rationals whose denominators are not divisible by primes outside S , and Z S ⁎ the multiplicative group of invertible elements ( S -units) in Z S . For a non-empty subset A of Z S , denote by G S ( A ) the graph with vertex set A and with an edge between a and b if and only if a − b ∈ Z S ⁎ . This type of graphs has been studied by many people. In the present paper we deal with the representability of finite (simple) graphs G as G S ( A ) . If A ′ = u A + a for some u ∈ Z S ⁎ and a ∈ Z S , then A and A ′ are called S -equivalent, since G S ( A ) and G S ( A ′ ) are isomorphic. We say that a finite graph G is representable/infinitely representable with S if G is isomorphic to G S ( A ) for some A /for infinitely many non- S -equivalent A . We prove among other things that for any finite graph G there exist infinitely many finite sets S of primes such that G can be represented with S . We deal with the infinite representability of finite graphs, in particular cycles and complete bipartite graphs. Further, we consider the triangles in G for a deeper analysis. Finally, we prove that G is representable with every S if and only if G is cubical. Besides combinatorial and numbertheoretical arguments, some deep Diophantine results concerning S -unit equations are used in our proofs. In Part II, we shall investigate these and similar problems over more general domains.

Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k ⩾ 4 and L ⩾ 3 there are only finitely many arithmetic progressions of the form (x0l0,x1l1,...,xk−1lk−1) with xi ∈ ℤ, gcd(x0, xl) = 1 and 2 ⩽ li ⩽ L for i = 0, 1, …, k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,…, 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [9], Darmon and Granville [6] as well as Chabautys method applied to superelliptic curves.

On the number of equivalence classes of binary forms of given degree and given discriminant

We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We have proved more general results for binary forms with coefficients in the ring of S-integers of a number field.

Perfect Powers in Products with Consecutive Terms from Arithmetic Progressions, II

There is an extensive literature on perfect powers and “almost” perfect powers in products of the form

Representation of finite graphs as difference graphs of S-units. II

n\left( {n + d} \right) \ldots \left( {n + \left( {k - 1} \right)d} \right)