It is shown that if s is large as a function of k and of
ε
>
0
\varepsilon > 0
, then the diophantine equation
a
1
x
1
k
+
⋯
+
a
s
x
s
k
=
b
1
y
1
k
+
⋯
+
b
s
y
s
k
{a_1}{x_1}^k + \cdots + {a_s}x_s^k = {b_1}y_1^k + \cdots + {b_s}y_s^k
with positive coefficients
a
1
,
…
,
a
s
{a_1}, \ldots ,{a_s}
,
b
1
,
…
,
b
s
{b_1}, \ldots ,{b_s}
has a nontrivial solution in nonnegative integers
x
1
,
…
,
x
s
{x_1}, \ldots ,{x_s}
,
y
1
,
…
,
y
s
{y_1}, \ldots ,{y_s}
not exceeding
m
(
1
/
k
)
+
ε
{m^{\left ( {1/k} \right ) + \varepsilon }}
, where m is the maximum of the coefficients.