= Unit equation over a number field
{title2=$\alpha u+\beta v=1,\qquad u,v\in\mathcal O_K^\times$}
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.
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