  We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F}
\psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is
the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$.
If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs
through $F^\times$, we always obtain at least three distinct values unless $d$
is degenerate (a power of the characteristic of $F$ modulo $|F^\times|$).
Choices of $F$ and $d$ for which we obtain only three values are quite rare and
desirable in a wide variety of applications. We show that if $F$ is a field of
order $3^n$ with $n$ odd, and $d=3^r+2$ with $4 r \equiv 1 \pmod{n}$, then
$W_{F,d}(a)$ assumes only the three values $0$ and $\pm 3^{(n+1)/2}$. This
proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The
proof employs diverse methods involving trilinear forms, counting points on
curves via multiplicative character sums, divisibility properties of Gauss
sums, and graph theory.
