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We know that infinitely many primes are the sums of two squares, but what about other powers than two? For what other powers, $k$, do we know the minimum number, $n$, such that infinitely many primes are the sum of $n$ $k$-th powers?

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    Thanks for the edit @projectilemotion. – 2017-01-05

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In general, every integer is the sum of finitely many $k$-th powers. For the minimum number see Waring's problem. For every $k$, let $g(k)$ denote the minimum number of $k$-th powers needed to represent all integers. For primes these minimal numbers are bounded, of course, by $g(k)$. We have $g(2)=4$, but infinitely many primes are the sum of $2$ squares. Also $g(3)=9$, but infinitely many primes conjecturally are the sum of three cubes (all primes not of the form $p\equiv \pm 4 \bmod 9$), see here. Unfortunately these minimal numbers are not known in general.

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    Many thanks for this, Dietrich. It would also be interesting if _g_ (_n_) for primes in particular were less than _g_ (_n_) for numbers in general, e.g. if _g_ (2) for primes were less than 4, or _g_ (3) for primes were less than 9. I take it, however, that there is no _n_ for which it is known that _g_ (_n_) for primes in particular is less than _g_ (_n_) for numbers in general, and I also take it that when we do know _g_ (_n_) for some _n_ we know that _g_ (_n_) for primes is the same as _g_ (_n_) for numbers in general? – 2017-01-04
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    $g(2)$, for example, cannot be less than $4$ for primes, because primes like $7$ are the sum of no less than $4$ squares. Also, the prime $23$ cannot be represented with less than $9$ (positive) cubes. So I suppose indeed that $g(n)$ is the same for primes as for all integers. – 2017-01-04
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    My thanks again, Dietrich. Indeed, I see that *all** the numbers that require 9 cubes (23 and 239) are prime. At least one of the numbers that requires 19 4th-powers is prime (79), though not all such are prime (e.g. 319). Further, one of the numbers (the only one?) that requires 37 5th-powers is 223, a prime. (Of course, some composite numbers such as 15 require four squares too.) – 2017-01-04