Suppose a curve of degree $d$ in the plane passes through $\frac{d^2+3d}{2}$ lattice points. Must it pass through another lattice point?



Must a curve containing $(d^2+3d)/2$ lattice points contain another?
Yes if $d=1$, but no for $d\ge 2$. 


[Schinzel's Theorem](http://mathworld.wolfram.com/CircleLatticePoints.html) says that for every $n\in \mathbb{N}$, there is a circle with exactly $n$ lattice points on the circumference. In particular, there is a circle passing through exactly $5$ lattice points, not $6$, which is a counterexample in degree $2$: 
$$\left(x-\frac{1}{3}\right)^2 +y^2 = \frac{5^4}{9}$$


Circles containing more points can be used to construct reducible counterexamples with higher degrees, as can curves like $xy=m$ where we can control the number of factors of $m$. 


