  Planar functions over finite fields give rise to finite projective planes and
other combinatorial objects. They were originally defined only in odd
characteristic, but recently Zhou introduced a definition in even
characteristic which yields similar applications. In this paper we show that
certain functions over $\mathbb{F}_{2^r}$ are planar, which proves a conjecture
of Schmidt and Zhou. The key to our proof is a new result about the
$\mathbb{F}_{q^3}$-rational points on the degree-$(q-1)$ Fermat curve
$x^{q-1}+y^{q-1}=z^{q-1}$.
