Description
Abstract:
Exponential Diophantine equations, say of the form
(1) u_1 + ... + u_k = b,
where the u_i are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the soultions only in case of k=2 (by results of Gyory and others), for k>2 only the number of so-called non-degenerate solutions can be bounded (by the Thue-Siegel-Roth-Schmidt method; see also results of Evertse and others). In particular, there is a big need for a method which is capable to solve (1) completely in concrete cases.
In the talk we present such a new, heuristic algorithm, ultimately based upon Skolem's conjecture (which roughly says that if (1) has no solutions, then it has no solutions modulo m with some m). We give several applications, as well. Then we provide a proof for Skolem's conjecture in some cases with k=2,3. (The handled cases include Catalan's equation and Fermat's equation,
too - the precise connection will be explained in the talk.) Note that previously Skolem's conjecture was proved only for k=1, by Schinzel.
For Zoom access please contact Andras Biro (biro.andras[a]renyi.hu).