In this paper, we discuss minimal free resolutions of the homogeneous ideals of quasi-complete intersection space curves. We show that if
X
X
is a quasi-complete intersection curve in
P
3
\mathbb P^3
, then
I
X
I_X
has a minimal free resolution
\[
0
→
⊕
i
=
1
μ
−
3
S
(
d
i
+
3
+
c
1
)
→
⊕
i
=
1
2
μ
−
4
S
(
−
e
i
)
→
⊕
i
=
1
μ
S
(
−
d
i
)
→
I
X
→
0
,
0\to \oplus _{i=1}^{\mu -3} S(d_{i+3}+c_1)\to \oplus _{i=1}^{2\mu -4}S(-e_i)\to \oplus _{i=1}^\mu S(-d_i)\to I_X\to 0,
\]
where
d
i
,
e
i
∈
Z
d_i,e_i\in \mathbb Z
and
c
1
=
−
d
1
−
d
2
−
d
3
c_1=-d_1-d_2-d_3
. Therefore the ranks of the first and the second syzygy modules are determined by the number of elements in a minimal generating set of
I
X
I_X
. Also we give a relation for the degrees of syzygy modules of
I
X
I_X
. Using this theorem, one can construct a smooth quasi-complete intersection curve
X
X
such that the number of minimal generators of
I
X
I_X
is
t
t
for any given positive integer
t
∈
Z
+
t\in \mathbb Z^+
.