For the ideal
p
\mathfrak {p}
in
k
[
x
,
y
,
z
]
k[x, y, z]
defining a space monomial curve, we show that
p
(
2
n
−
1
)
⊆
m
p
n
\mathfrak {p}^{(2 n - 1)} \subseteq \mathfrak {m} \mathfrak {p}^{n}
for some positive integer
n
n
, where
m
\mathfrak {m}
is the maximal ideal
(
x
,
y
,
z
)
(x, y, z)
. Moreover, the smallest such
n
n
is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that
p
(
3
)
⊆
m
p
2
\mathfrak {p}^{(3)} \subseteq \mathfrak {m} \mathfrak {p}^2
if
k
k
is a field of characteristic not
3
3
; however, the stable Harbourne conjecture holds for space monomial curves as they claimed.