Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 20G / d / Solution

Letwith $r$ entries equal to one and $s$ entries equal to two. Its coordinate sum isso $e\notin H$. Since the fundamental-unit logarithms form a basis of the codimension-one hyperplane $H$, the vectorsform a basis of ${\mathbb{R}}^{r+s}$. This gives unique real numbers $t,{\lambda }_i$ such that

The sum of the coordinates on the left isEvery unit logarithm has coordinate sum zero, so summing the right side gives $t(r+2s)$. Therefore