Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares $n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer coefficients (possibly square) $f(x)$ maximizing the number of solutions to $f(x)=u_i$.
Solutions are integral points on the elliptic curve $n=y^2-f(x)^2$.
Unless one can use the group law and scale rationals (related to fourth power free), boundedness is open problem, so I am interested in explicit records.
In terms of rank, my current record is $37$ points (not counting the sign of $x$) on $ y^2-(x^2-565215^2)^2=40224510201185827416900$ of rank $9$.
This was found by expressing $u_i=u_i'^2-v_i'^2$ and checking for the most common $v_i'$.
Also, what do heuristic say about this record(s)?