I am reading a survey Reciprocity laws and Galois representations: recent breakthroughs. On page 5, it mentions that
Theorem 2.1.5 shows that a prime $p \neq 2, 5$ divides an integer of the form $x^2+5$ if and only if $p$ satisfies a congruence condition modulo $20$, which happens to be the condition that $p\equiv 1,3,7,9$ (mod $20$). But such a prime (for instance $7$) is not necessarily of the form $a^2 + 5b^2$. It turns out that Fermat’s method of descent fails in this context. Phrased in modern terms, the culprit is the failure of $\mathbb{Z}[\sqrt{-5}]$ to have the property of unique factorization into primes.
My question is: what is the detail of the last sentence in the quotation of the survey? What is the relationship between Fermat's method of descent and UFD?
The survey mentions little about Fermat's method of descent (Actually it only mentions it on the top of page 3, without any detail). I search for the Number Theory: An approach through history from Hammurapi to Legendre by André Weil for some details of Fermat's method of descent. In the section VIII of Chapter 2 of Weil's book, it proves a lemma by Euler:
Lemma 2. For any $N = a^2 + b^2$, let $q = x^2 + y^2$ be a prime divisor of $N$. Then $N/q$ has a representation $u^2 + v^2$ such that the representation $N = a^2 + b^2$ is one of those derived by composition from it and from $q = x^2 + y^2$.
Then it says that
If now, as before, $p$ is the greatest prime divisor of $N = a^2 + b^2$, with $a, b$ mutually prime, and all prime divisors of $N$ except $p$ are known to be of the form $x^2 + y^2$, we can apply lemma 2 to any one of these, say $q$, then to any prime divisor other than $p$ of $N /q$, etc., until we obtain for $p$ itself an expression as a sum of two squares. This completes the proof of Fermat's statement, as obtained by Euler more than a century later. Except for minor details, Fermat's proof could have been much the same.
I think Fermat's method of descent mentioned in the survey is just like the one mentioned by Weil. The quotation part of the survey may apply a generalization of this: if prime $q$ dividing $N=x^2+ny^2$ with $x,y,n$ coprime to $q$, then $q$ has a representation $u^2+nv^2$ for some special $n$ and the survey claims that the generalization may fail when $\mathbb{Z}[\sqrt{-n}]$ is not a UFD. When $n<4$ we can assume that $q$ is the greatest prime divisor of $N$, so we can apply the procedure mentioned by Weil. But I cannot work out how this relates with UFD.
Moreover, note that $\mathbb{Z}[\sqrt{-3}]$ is not a UFD, but the descent method success when $n=3$. When $n=3$, $\mathbb{Z}[\omega]$ is a UFD, where $\omega=\frac{-1+\sqrt{-3}}{2}$. So we may substitute the $\mathbb{Z}[\sqrt{-n}]$ by the algebraic integer ring $O_{\mathbb{Q}(\sqrt{-n})}$?
I know that we can prove a version of lemma 2 when a prime $q=x^2+ny^2$ divides a number $N=a^2+nb^2$ as long as $\mathbb{Z}[\sqrt{-n}]$ is a UFD, but this lemma actually works for any positive integer $n$, so this may not be a satisfying explain.