Max A. Alekseyev, Department of Mathematics, George Washington University

Computational methods for solving exponential and exponential-polynomial Diophantine equations
We present computational methods for solving Diophantine equations of two forms:

A*P^m + B*Q^n +C*R^k = 0 and A*m^2 + B*m + C = D*Q^n,

where $A,B,C,D,P,Q,R$ are given integers, $P,Q,R>0$, and $m,n,k$ are unknowns. The methods are based on modular arithmetics and aimed at bounding values of the variables, which further enables solving the equations by exhaustive search. We illustrate the methods workflow with some Diophantine equations like 4 + 3^n - 7^k = 0 and 2*m^2 + 1 = 3^n and show how one can compute all their solutions.