TITLE: A conjecture by Euler about $8n+3$
QUESTION [12 upvotes]: Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum 
$$8n+3=(2k-1)^2+2p,$$
where $k$ is a positive integer, and $p$ is a prime.
I want to know whether there has been progress  on the problem. Could you recommend some references? Thanks a lot.

REPLY [16 votes]: Barry Mazur discussed this problem in his presentation of January 2012, Why is it plausible? Mazur says there that it is still unsettled.