A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

Linear forms in logarithms and exponential Diophantine equations

International audience This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

Exponential diophantine equations in rings of positive characteristic

In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations [Formula: see text] where [Formula: see text] are constants from matrix ring of characteristic [Formula: see text], [Formula: see text] are indeterminates. For any solution [Formula: see text] of the system we construct a word (over an alphabet containing [Formula: see text] symbols) [Formula: see text] where [Formula: see text] is a [Formula: see text]-tuple [Formula: see text], [Formula: see text] is the [Formula: see text]th digit in the [Formula: see text]-adic representation of [Formula: see text]. The main result of this paper is following: the set of words corresponding in this sense to solutions of a system of exponential-Diophantine equations is a regular language (i.e., recognizable by a finite automaton). There exists an algorithm which calculates this language. This algorithm is constructed in the paper.