Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number?  

It *seems* that for every odd $m$ there are many $(a,b)\in \mathbb{N}^2$ which sum to $m$ and their sum of squares give a prime number but I don't see how to prove this.  

Or, equivalently, is it true that if we list all primes $p=a^2+b^2\equiv 1\pmod{4}$ then $a+b$ covers every odd integer? 



Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime
The question is not new. It is originally due to Ming-Zhi Zhang. One may consult <https://oeis.org/A036468> .


