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?
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$\begingroup$ This seems to be some kind of strengthening of Bertrand`s postulate. $\endgroup$ – Shalom Mar 27 '18 at 20:39

1$\begingroup$ Can you generalize this to cubes and higher powers? $\endgroup$ – Shalom Mar 27 '18 at 20:44

1$\begingroup$ You're asking whether for every such $m$, the polynomial $a^2+(ma)^2 = 2a^22am+m^2$ takes a prime value for some $1\le a<m$. This is a Goldbachlike variation of the question "does an irreducibile quadratic polynomial take infinitely many prime values", in the same way that the Goldbach problem is a variation of the twin primes conjecture. Since we don't know how to establish the quadratic version even for a fixed quadratic polynomial, this is probably a hard problem. $\endgroup$ – Greg Martin Mar 27 '18 at 21:09

$\begingroup$ @GregMartin I also suppose this is a hard problem, but (maybe) it is not as hard as the "infinitely many primes in a quadratic polynomial". $\endgroup$ – Konstantinos Gaitanas Mar 27 '18 at 21:16
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The question is not new. It is originally due to MingZhi Zhang. One may consult https://oeis.org/A036468 .

$\begingroup$ Do You prove that @ZhiWei Sun, or this is conjecture? $\endgroup$ – Đào Thanh Oai Jul 1 '18 at 10:14