Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields

Mateo Matijasevick, Santiago Rodríguez, Gregorio Salazar

Let $k$ be a positive integer, and consider the sequences of positive rationals with $x_0\in \mathbb{N}$ and $x_{n+1}=k/(x_0+x_1+\cdots +x_n)$. Write $x_n=a_n/b_n$ in lowest terms. We show th…