2025/09/22 by Kun Yi Ong, Ong, Kun Yi, Eddie Shahril Ismail +1
Computer Science · #11D09 #11D45 #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2509.17882
openalex publication_date 2025/09/22 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
The Pell equation x2 - Dy2 = 1 with non-square D > 1 has infinitely many integer solutions, yet most research has centered on the asymptotic behavior of fundamental units as D varies. By contrast, the exact distribution of solutions for a fixed D within bounded regions has received little attention. In this paper, we contribute to this direction by giving an explicit enumeration of all solutions to the Pell equation inside the square |x| + |y| ≤ λ for any λ> 0. We further extend our results to the shifted Pell equation (x-a)2 - D(y-b)2 = 1 for integers a and b, obtaining exact counts for sufficiently large λ.