Clemens Fuchs

Complete Solution of a Problem of Diophantus and Euler

It is proved that there does not exist a set of four positive integers with the property that the product of any two of its distinct elements plus their sum is a perfect square. This settles an old problem investigated by Diophantus and Euler.

Composite rational functions expressible with few terms

We consider a rational function f which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number l of terms. Then we look at the possible decompositions f(x) = g(h(x)), where g, h are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of g is bounded only in terms of l (and we provide explicit bounds). This supports and quantifies the intuitive expectation that rational operations of large degree tend to destroy lacunarity. As an application in the context of algebraic dynamics, we show that the minimum number of terms necessary to express an iterate h of a rational function h, tends to infinity with n, provided h(x) is not of an explicitly described special shape. The conclusions extend some previous results for the case when f is a Laurent-polynomial; the proofs present several features which did not appear at all in the special cases treated so far.

Diophantine m-tuples for linear polynomials

In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients.

Composite rational functions having a bounded number of zeros and poles

In this paper we study composite rational functions which have at most a given number of distinct zeros and poles. A complete algorithmic characterization of all such functions and decompositions is given. This can be seen as a multiplicative analog of a result due to Zannier on polynomials that are lacunary in the sense that they have a bounded number of non-constant terms.

A polynomial variant of a problem of Diophantus for pure powers

In this paper, we prove that there does not exist a set of 11 polynomials with coefficients in a field of characteristic 0, not all constant, with the property that the product of any two distinct elements plus 1 is a perfect square. Moreover, we prove that there does not exist a set of 5 polynomials with the property that the product of any two distinct elements plus 1 is a perfect kth power with k ≥ 7. Combining these results, we get an absolute upper bound for the size of a set with the property that the product of any two elements plus 1 is a pure power.

Only finitely many Tribonacci Diophantine triples exist

Abstract Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1 ≤ u < v < w such that uv + 1, uw + 1, vw + 1 are Tribonacci numbers. The proof depends on the Subspace theorem.

On the Diophantine equation Gn(x) = Gm(P(x)): Higher-order recurrences

Let K be a field of characteristic 0 and let (G n (x))∞n=0 be a linear recurring sequence of degree d in K[x] defined by the initial terms G 0 ,..., C d-1 ∈ K[x] and by the difference equation G n+d (x) = A d-1 (x)G n+d-1 (x) + ... + A 0 (x)G n (x), for n > 0, with A 0 ,..., A d-1 ∈ K[x]. Finally, let P(x) be an element of K[x]. In this paper we are giving fairly general conditions depending only on G 0 ,..,G d-1 , on P, and on A 0 ,...,A d-1 under which the Diophantine equation G n (x) = G m (P(x)) has only finitely many solutions (n, m) ∈ Z 2 , n, m > 0. Moreover, we are giving an upper bound for the number of solutions, which depends only on d. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

Commutative algebraic groups and p-adic linear forms

Let

Diophantine Triples and k-Generalized Fibonacci Sequences

G

30 years of collaboration

be a commutative algebraic group defined over a number field