Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 166 / 2 / h

For an integer $m>1$, let ${\alpha }_m$ be a root ofIts roots areThe polynomial is irreducible over $\mathbb{Q}$: its discriminant is $4m(m-1)$, and the product of the coprime consecutive integers $m$ and $m-1$ cannot be a square unless both are squares, which is impossible for consecutive positive squares beyond $0,1$.

The height-Mahler measure formula givesBoth algebraic conjugates of ${\alpha }_m+1$ exceed $1$, soConsequentlyFix, for example, $\delta =1/10$. For all sufficiently large $m$, the ratio is larger than $1+\delta$ by a fixed margin, while $H({\alpha }_m)\to \infty$. Hence, for every $C>0$, some sufficiently large $m$ satisfies

Solved by gpt-5.6-sol high.