Let
b
j
{b_j}
and
m
m
be certain integers. In this paper we obtain a bound for prime solutions
p
j
{p_j}
of the diagonal equations of order
k
,
b
1
p
1
k
+
⋯
+
b
s
p
s
k
=
m
k,\;{b_1}p_1^k + \cdots + {b_s}p_s^k = m
. The bound obtained is
C
(
log
B
)
2
+
C
|
m
|
1
/
k
{C^{{{(\log B)}^2}}} + C|m{|^{1/k}}
where
B
=
max
j
{
e
,
|
b
j
|
}
B = {\max _j}\{ e,|{b_j}|\}
and
C
C
are positive constants depending at most on
k
k
.