# Even numbers has the form $a+b$ where $\frac{a^2+b^2}{2}$ is prime

Conjecture:
All even numbers $n>2$ can be written $n=a+b$ where $a,b\in\mathbb N^+$ and $\frac{a^2+b^2}{2}\in\mathbb P$.

Is tested for $n<10,000,000$.

This conjecture is related to and maybe dependent on Any odd number is of form $a+b$ where $a^2+b^2$ is prime.

• Equivalently, for every even $n=2k$, there is $a<n$ such that $\dfrac{2a^2-2an+n^2}2 = a^2-2ak +2k^2 \in \Bbb P$ – Kenny Lau Aug 30 '17 at 5:44
• Equivalently, for every $k>1$ there is $a<2k$ such that $(a-k)^2+k^2\in\mathbb{P}$. Then without loss of generality we may restrict to $a<k$, and the question comes down to asking whether there is a prime in the sequence $\{k^2+c^2:\ 0<c<k\}$ for every $k>1$. – Servaes Aug 30 '17 at 5:50
• Isn't this (like the linked question) a special case of the Bateman-Horn conjecture? – Hagen von Eitzen Aug 30 '17 at 6:37
• @HagenvonEitzen: I can't see that. – Lehs Aug 30 '17 at 12:47