Pell Equations and ℱpl‐Continued Fractions

Abstract

In this note, the solvability of the Pell equation, X 2 − D Y 2 = 1 , is discussed over ℤ × p l ℤ . In particular, we show that this equation is solvable over ℤ × p l ℤ for each prime p and natural number l . Moreover, we show that solutions to the Pell equation over ℤ × p l ℤ are completely determined by the ℱ p l -continued fraction expansion of D .