Abstract:

Given integers $k\ge 2$ and $\ell\ge 3$, let $S_{k,\ell}^*$ stand for the set of those positive integers $n$ which can be written as $n=p_1^k+p_2^k+\ldots+p_\ell^k$, where $p_1,p_2,\ldots,p_\ell$ are distinct prime factors of $n$. We study the properties of the sets $S^*_{k,\ell}$ and we show in particular that, given any odd $\ell\ge 3$, $${\char93 \bigcup_{k=2}^\infty S_{k,\ell}^*=+\infty}$$.