Why are there only 436 numbers not of the form $x^2+My^2$ for $x>0$, $y>1$ and $M>0$? This is A074885 from OEIS. The last number is 1875902.
Can the following argument be fixed up?
I expect $N$ to be a quadratic residue $\mod p$
for about half the primes $p$ that are not factors of $N$.
Let $y=p$; there will then be an $0<x<p^2/2$ with $x^2=N \mod p^2$.
$M$ will then be a positive integer if $N>p^4/4$.
So there is a solution for $x$, $y=p$ and $M$ if $N$ is a quadratic residue for some $p<(4N)^{1/4}$. With $O(4(4N)^{1/4}/\log 4N)$ primes $p$ in that range, the chance of $N$ being non-quadratic residue for all primes in that range would be
$$2^{-4(4N)^{1/4}/\log 4N}$$
That gets exponentially small, so the no-solutions would likely run out, but only once
$(4N)^{1/4}$ gets large compared with $\log^2 N$.
It doesn't say there are only $436$ such numbers; it says one guy found $436$ such numbers, and thinks there aren't any more. Also, his work seems to have been purely computational, not at all theoretical, which means nobody knows why there are only $436$ such numbers (if, indeed, there are only $436$ such numbers).