The Number of Solutions of Pell Equations \( x^2 – ky^2 = N\) And \( x^2 + 2y – ky^2 = N\) Over \(\mathbb{F}_p\)

Abstract

Let \(p\) be a prime number such that \(p \equiv 1, 3 \pmod{4}\), let \(\mathbb{F}_p\) be a finite field, and let \(N \in \mathbb{F}_p^* = \mathbb{F}_p – \{0\}\) be a fixed element. Let \(P_p^k(N): x^2 – ky^2 = N\) and \(\tilde{P}_p^k(N): x^2 + 2y – ky^2 = N\) be two Pell equations over \(\mathbb{F}_p\), where \(k = \frac{p-1}{4}\) or \(k = \frac{p-3}{4}\), respectively. Let \(P_p^k(N)(\mathbb{F}_p)\) and \(\tilde{P}_p^k(N)(\mathbb{F}_p)\) denote the set of integer solutions of the Pell equations \(P_p^k(N)\) and \(\tilde{P}_p^k(N)\), respectively. In the first section, we give some preliminaries from the general Pell equation \(x^2 – ky^2 = \pm N\). In the second section, we determine the number of integer solutions of \(P_p^k(N)\). We prove that \(P_p^k(N)(\mathbb{F}_p) = p+1\) if \(p \equiv 1 \pmod{4}\) or \(p \equiv 7 \pmod{12}\) and \(P_p^k(N)(\mathbb{F}_p) = p-1\) if \(p \equiv 11 \pmod{12}\). In the third section, we consider the Pell equation \(\tilde{P}_p^k(N)\). We prove that \(\tilde{P}_p^k(N)(\mathbb{F}_p) = 2p\) if \(p \equiv 1 \pmod{4}\) and \(N \in Q_p\); \(\tilde{P}_p^k(N)(\mathbb{F}_p) = 0\) if \(p \equiv 1 \pmod{4}\) and \(N \notin Q_p\); \(\tilde{P}_p^k(N)(\mathbb{F}_p) = p+1\) if \(p \equiv 3 \pmod{4}\).