Some congruence properties of the Pell equation (Q2888161)

The group \(G(N)\) of the integer solutions \((x,y)\) of the Pell equation \(x^2-Ny^2=1\) can be viewed as a subgroup of \(\mathrm{GL}_2(\mathbb{Z})\). For an integer \(m>1\), we have the reduction group homomorphism \(\text{red}_m:G(N) \to \mathrm{GL}_2(\mathbb{Z}/m\mathbb{Z})\). In this paper, the authors study properties of the function \(g_N(m)=|\text{red}_m(G(N))|\). They show that if an odd prime \(p\) divides \(N\), then \(g_N(p) | 2p\), and if \(p\) does not divide \(N\), then \(g_N(p)\) divides \(p-1\) or \(p+1\).