We prove that if
x
m
+
a
x
n
x^m + ax^n
permutes the prime field
F
p
\mathbb {F}_p
, where
m
>
n
>
0
m>n>0
and
a
∈
F
p
∗
a\in \mathbb {F}_p^*
, then
gcd
(
m
−
n
,
p
−
1
)
>
p
−
1
\gcd (m-n,p-1) > \sqrt {p}-1
. Conversely, we prove that if
q
≥
4
q\ge 4
and
m
>
n
>
0
m>n>0
are fixed and satisfy
gcd
(
m
−
n
,
q
−
1
)
>
2
q
(
log
log
q
)
/
log
q
\gcd (m-n,q-1) > 2q(\log \log q)/\log q
, then there exist permutation binomials over
F
q
\mathbb {F}_q
of the form
x
m
+
a
x
n
x^m + ax^n
if and only if
gcd
(
m
,
n
,
q
−
1
)
=
1
\gcd (m,n,q-1) = 1
.