On certain plane curves with many integral points

Abstract

We define a sequence of polynomials $P_d \in \funnyZ[x,y]$, such that $P_d$ is absolutely irreducible, of degree d, has low height, and has at least $d^2+2d+3$ integral solutions to $P_d(x,y)=0$. We know of no other nontrivial family of polynomials of increasing degree with as many integral solutions in terms of their degree.