Unit equation over a number field ($\alpha u+\beta v=1,u,v\in {\mathcal{O}}_K^{\times }$)

For fixed nonzero $\alpha ,\beta$ in a number field, this equation has finitely many unit solutions. By the Dirichlet unit theorem, a large unit-exponent vector makes one unit exponentially small at some archimedean embedding. The equation makes the other fixed multiple exponentially close to one. Its principal logarithm is a nonzero linear form in fixed algebraic logarithms, with coefficients bounded by the exponent size; a polynomial lower bound contradicts the exponential upper bound. Finite torsion and finitely many embeddings allow uniform constants.