It has been shown, by elementary methods, that every positive integer can be expressed as the sum of $4$ squares. This type of result has been proven for many different powers $p$, for example, when $p=5$,
$$
\forall n \in \Bbb{N}: \exists \{x\_i\} x\_i \in \Bbb{N}^{37}, 1 \leq i \leq 37 : n = \sum x\_i^5
$$ 
(and the only number requiring the sum of $37$ fifth powers happens to be 223).


Here, $\Bbb{N}$ does include zero and repetition is allowed, so for instance $6 = 2^2 + 1^2 + 1^2 + 0^2$. 


However easy or difficult it may be to find the minimal $k$ with this property for any given $p$, the conjecture that started this sort of questioning (due, I believe, to Waring at Cambridge in 1770), is that:


(W) For any $p \in \Bbb{Z}^+$ there exists *some* $k\in \Bbb{Z}^+$, which I will for now fcall $k(p)$, satisfying 
$$
\forall n \in \Bbb{N}: \exists \{x\_i\} x\_i \in \Bbb{N}^{k(p)}, 1 \leq i \leq k(p) : n = \sum\_{i=1}^{k(p)} x\_i^p
$$ 


Thus if this conjecture (W) is known to be true, then one knows that for $p=5$ some finite number of terms in the some suffices to express all integers, although you would not know that $k(5) = 37$.


The tools needed to prove (W) are quite different than the tools needed to find the minimal $k(p)$ for a specified $p$.


My question is whether (W) has been proven, and/or whether there are results proving the condition in (W) for some infinite set of values of $p$.



Is it proved that for every integer $p>0$ there exists an integer $k>0$ such that every integer $n>0$ can be expressed as $j_1^p+\dots+j_k^p$?
You ask if Waring's conjecture has been proved. The answer is yes (Hilbert 1909), and you can read about its history [here](https://en.wikipedia.org/wiki/Waring's_problem). 


Your $k(p)$ is usually denoted by $g(p)$, and its minimal value is almost precisely known. It is harder to estimate the related quantity $G(p)$ that is meant to work for all sufficiently large $n$'s only, and you can read about the state-of-the-art at the linked page.


If you are interested in this topic, I recommend you Vaughan's book "The Hardy-Littlewood method".


