Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue’s Lemma.
What is known about sums of $n$ higher powers resulting in primes?
I tried $a^3+b^3+c^3$, asked Wolfram and found $3,17,29,43,73$, a sequence that I don’t know (EDIT: thanks to Matthew: A007490). Interestingly, that $251$ has even $2$ decompositions, $1^3+5^3+5^3=2^3+3^3+6^3$.
Can anybody help here?
Any cube is congruent to $0$, $1$, or $-1$ modulo $9$. It follows that the sum of three cubes cannot be congruent to $4$ or $5$ modulo $9$. So we have a congruential restriction analogous to the case of two squares.
By Dirichlet’s Theorem on primes in arithmetic progressions (or undoubtedly by much more elementary means) one can show that there are infinitely many primes congruent to $4$ modulo $9$, and also infinitely many primes congruent to $5$. Thus there are infinitely many primes that cannot be represented as the sum of three cubes.
About primes not congruent to $4$ or $5$ modulo $9$, one cannot say much. A long-standing conjecture is that every integer which is not congruent to $4$ or $5$ modulo $9$ is the sum of $3$ cubes. (Here negative cubes are allowed.) There has been a lot of computation on this problem.
For other powers, one should go to the vast literature on Waring’s Problem.