In this paper, the existence of effectively computable bounds on the solutions to the diophantine equation apx+bqy = c+dpzqw is shown. In this equation p, q are taken to be fixed relatively prime positive integers and a, b, c, d positive integers. The methods involve the application of linear forms in both real and p-adic logarithms. Also, a result on an inequality involving S-integers is used. All constants involved can be explicitly computed in terms of the parameters a, b, c, d, p, q, conceivably allowing one to list all solutions to (*) for any set of parameters. It is also indicated how the bounds in a particular case can be reduced to allow the practical solution of the equation. Finally, the methods are demonstrated by the solving of the equations 2x + 3y = 1 + 2z3w and 5 · 2x + 7 · 3y = 11 + 2z3w.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory