A Neighbouring-Denominator Case of the Erd\H{o}s--Mahler Conjecture

Abstract

In 1939, Erd\H{o}s and Mahler conjectured that an irrational real number $\xi$ must be a Liouville number whenever $P(p_nq_n)$ is bounded for infinitely many convergents $p_n/q_n$, where $P(N)$ denotes the largest prime factor of a nonzero integer $N$. In this note, we prove a neighbouring-denominator case of their conjecture: if $P(p_nq_nq_{n+1})$ is bounded for infinitely many $n$, then $\xi$ is a Liouville number. The proof combines the determinant identity for consecutive convergents with a fixed-base estimate for linear forms in $p$-adic logarithms.