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...