Suppose, $n\ge 24$ is a natual number. Then, $n$ can be written as a sum of squares of primes (the squares of the primes need not be distinct) because $n$ can always be written in the form $4a+9b$ with non-negative integers $a,b$. So, we can find primes (not necessarily distinct) such that $$(1)\ \ \ \ p_1^2+p_2^2+\cdots+p_k^2=n$$
But is there a positive integer $k$, such that every positive integer $n\ge16$ can be written as a sum of at most $k$ squares of a prime ? In other words, a number $k$ , such that $(1)$ has always a solution ?