On a problem of Erdős and Graham

In this paper we provide bounds for the size of the solutions of the Diophantine equation

$$\begin{aligned} x(x+1)(x+2)(x+3)(x+m)(x+m+1)(x+m+2)(x+m+3)=y^2, \end{aligned}$$

where \(4\le m\in \mathbb {N}\) is a parameter. We also determine all integral solutions for \(1\le m\le 10^6.\)

     

Markoff–Rosenberger triples and generalized Lucas sequences

Periodica Mathematica Hungarica - We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ with $$(x,y,z)=(U_i,U_j,U_k)$$ , where $$U_i$$ denotes the...     

Finiteness results for F-Diophantine sets

Monatshefte für Mathematik - Diophantine sets, i.e. sets of positive integers A with the property that the product of any two distinct elements of A increased by 1 is a perfect square, have a...     

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 with x_i , gcd(x₀, x_l) = 1 and 2 l_i 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 Chabauty's method applied to superelliptic curves.