MathDB

Let

q>3

be a rational number, such that

q^2-4

is a perfect square of a rational number. The sequence

a_0, a_1, \ldots

is defined by

a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}

for all

i \geq 1

. Is it true that there exist a positive integer

n

and nonzero integers

b_0, b_1, \ldots, b_n

with sum zero, such that if

\sum_{i=0}^{n} a_ib_i=\frac{A} {B}

for

(A, B)=1

, then

A

is squarefree?

Problem Statement