A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation $${{5}^{u}}{{x}^{n}}-{{2}^{r}}{{3}^{5}}{{y}^{n}}=\pm 1,$$ in non-zero integers $x,y$ and positive integers $u,r,s$ and $n\ge 3$ . Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.