Abstract
Let
Q
(
x
)
=
Q
(
x
1
,
x
2
,
⋯
,
x
n
)
be a nonsingular quadratic form over
Z
, and
p
be an odd prime. A solution of the congruence
Q
(
x
)
≡
0
(
mod
p
m
)
is said to be a primitive solution if
p
∤
x
i
for some
i
. We prove that if
p
>
A
,
where
A
=
2
2
(
n
+
1
)
/
(
n
-
2
)
3
2
/
(
n
-
2
)
, then this congruence has a primitive solution, with
x
≤
6
1
/
n
p
(
m
/
2
)
+
(
m
/
n
)
whenever
n
>
m
and
m
≥
2
,
for every even
n
.