Class numbers and integer points on some Pellian surfaces

TL;DR

This paper estimates the number of integer points on Pellian surfaces within bounded regions, provides bounds on fundamental solutions for most cases, and relates these to class numbers, assuming a recent conjecture.

Contribution

It offers new bounds on integer points and fundamental solutions on Pellian surfaces, connecting these to class numbers under a conjecture.

Findings

Estimated integer points on Pellian surfaces in bounded regions

Lower bounds on fundamental solutions for most d in a class

Upper bounds on average class numbers assuming a conjecture

Abstract

We provide an estimate for the number of nontrivial integer points on the Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on the size of fundamental solutions for almost all $d$ in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.