nt.number theory - Sequence of Diophantine Equations

Is there some (huge) positive integer $M$ with the following property:
for any $z>M$, there exist positive integers $x, y_{1}, y_{2},..., y_{z}$
such that $x^x$ $=$ $y_{1}^{y_{1}}$+ $y_{2}^{y_{2}}$+ ... +$y_{z}^{y_{z}}$
?

[Please remark that the $y$'s are $geq$ $1$ and need not
to be necessarily distinct.]

As a [rather naive] way to attack this problem [which may (perhaps) be
related to some works of Robinson, Matiasevich, M. Davis, and Chao-Ko],
I'm thinking about lots of $1$'s, lots of $2$'s, and lots of
$(x-1)$'s.
Also, let us observe that, if $z$ has this property and $y_{i}$ $=$ $2$
for some $i$, then $z+3$ has the same property, too...