We explicitly solve a collection of binomial Thue equations with unknown degree and unknown $S$-unit coefficients, for a number of sets $S$ of small cardinality. Equivalently, we characterize integers $x$ such that the polynomial $x^2+x$ assumes perfect power values, modulo $S$-units. These results are proved through a combination of techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, and computational approaches to Thue equations of low degree. Along the way, we derive some new results on Fermat-type ternary equations, combining classical cyclotomy with Frey curve techniques.
