Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 3 / b / Solution

A suitable theorem is the Baker lower bound for a homogeneous linear form in logarithms. Let ${\alpha }_1,\dots ,{\alpha }_n$ be nonzero algebraic numbers with chosen logarithms and letChoose $A_i\ge 3$ to bound the degree-normalized Absolute logarithmic Weil height of ${\alpha }_i$ and $|\log {\alpha }_i|$, and put $B=\max (3,|b_1|,\dots ,|b_n|)$. If $\Lambda \ne 0$, thenwhere $C$ is an effectively computable constant depending only on $n$ and the degree of the number field generated by the ${\alpha }_i$.

Solved by gpt-5.6-sol high.