Diophantine equations with three monomials (Q6046937)

In this paper, a general algorithm for solving all three-monomial equations in two variables is presented. It is relied on deep classical results in the theory of two-variable Diophantine equations, such as Baker's resolution of superelliptic equations and Walsh's effective version of Runge's theorem, but is otherwise elementary. An interesting special case is the family of equations \(x^4 + axy + y^3 = 0\), where \(a\) is an integer. A table with the non-trivial solutions of these equations with \(1\leq a \leq 100\) is given. Furthermore, a method that reduces an arbitrary three-monomial equation, in any number of variables, to a finite number of equations that have three monomials in disjoint variables, is described. In many examples, the resulting equations are either easier than the original one, or are already solved in the literature. Two interesting examples are given: a family of equations that the proposed method reduces to Fermat's Last Theorem, and the description of all integer solutions to the equation \(ax^n+by^m= cz^k\), provided that at least one of the exponents \(n, m, k\) is coprime with the other two. Tables with cubic and quartic three monomials equations are given which are solvable and non solvable by the proposed method. Moreover, a simple sufficient condition is provided that guarantees that a given three-monomial equation is solvable by the presented method, and empirically check that the percentage of three-monomial equations satisfying this condition approaches 100\% when the number of variables goes to infinity.