Computational experiments suggests the conjecture:
All big enough odd numbers $N$ is of the form $N=\sum_{k=1}^n m_k$, where $\sum_{k=1}^n m_k^2$ is prime and all $m_k\in\mathbb Z^+$.
Since the possible combinations of partitions of $N$ increase with increasing $N$ and $n$, the case with $n=2$ is tested for $N<10^8$ and seems to be supported by heuristics, and the cases $n=3,4,5$ are tested for $N$ up to $200$, I'm interested to understand the following, which is my question:
Are there reasons to believe that a proof for some "big" $n$ with $N$ "big enough" would be within reach while proofs for "small" $n$ isn't, at the moment?