Let
p
(
n
)
p(n)
denote the number of partitions of
n
n
. Let
A
,
B
∈
N
A,B\in \mathbb {N}
with
A
>
B
A>B
and
ℓ
≥
5
\ell \geq 5
a prime, such that
\[
p
(
A
n
+
B
)
≡
0
(
mod
ℓ
)
,
n
∈
N
.
p(An+B)\equiv 0\pmod {\ell }, \quad n\in \mathbb {N}.
\]
Then we will prove that
ℓ
|
A
\ell |A
and
(
24
B
−
1
ℓ
)
≠
(
−
1
ℓ
)
\left (\frac {24B-1}{\ell }\right ) \neq \left (\frac {-1}{\ell }\right )
. This settles an open problem by Scott Ahlgren and Ken Ono. Our proof is based on results by Deligne and Rapoport.