Show that $(x + y\sqrt{-5})$ must be a prime in $\mathbb{Z}[\sqrt{-5}]$ Got these problems as separate sections of a question in a book's chapter on 'Divisibility & primes'.

  
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*Show that if $x^2 + 5y^2 =1$, then $x = \pm 1$.
  

Can state it in terms of two factors as : $(x + y\sqrt{-5})(x-y\sqrt{-5}) =x^2 + 5y^2$, with 
(i) $(x + y\sqrt{-5}) = 1$, (ii) $(x - y\sqrt{-5}) = 1$
Adding both (i) & (ii), get: $x = 1$, Subtracting (ii) from (i), get: $(y\sqrt{-5}) = 0$.
Unable to pursue after that, as $x=-1$ is not possible.


  
*Show that $(x + y\sqrt{-5})$ must be a prime in $\mathbb{Z}[\sqrt{-5}]$
  

The hint given is to use the unique factorization theorem for the integers, by supposing $(x + y\sqrt{-5}) = (a + b\sqrt{-5})(c + d\sqrt{-5})$. The hint asks to show that: $(x^2 + 5y^2) = (a^2 + 5b^2)(c^2 + 5d^2)$. I need some more hint or help to pursue, as squaring $(x + y\sqrt{-5}) = (a + b\sqrt{-5})(c + d\sqrt{-5})$ does not lead to $(x^2 + 5y^2) = (a^2 + 5b^2)(c^2 + 5d^2)$. My attempt is stated for squaring both sides below:L.H.S.: $(x + y\sqrt{-5})(x + y\sqrt{-5}) => x^2 + 2xy(-5) -5y^2$  R.H.S.: $(a + b\sqrt{-5})^2(c + d\sqrt{-5})^2 => (a + b\sqrt{-5})(a + b\sqrt{-5})(c + d\sqrt{-5})(c + d\sqrt{-5}) => (a^2 -5b^2 +2ab\sqrt{-5})(c^2 -5d^2 +2cd\sqrt{-5})$


  
*Find all primes less than 50 in integers that can be written in the form $x^2 + 5y^2$.
  

No clue except to first find the primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$.
Then trying to see if the factorization works, so starting with $2 = x^2 + 5y^2$, but cannot think further. Do I need to have $y$ as a imaginary number only, as $\sqrt{-5}$, or anything will work.

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*You're overthinking this one. We have two integers, $x$ and $y$, with $x^2 + 5y^2=1$. If $y$ is anything other than $0$, then $5y^2$ is greater than $1$, and since $x^2$ is non-negative, the equation is impossible. Having decided $y=0$, we are left with $x^2=1$, which has two solutions: $x=\pm 1$

*This question seems to be missing something. It is not true that every number of the form $x+y\sqrt{-5}$ is prime in $\Bbb Z[\sqrt{-5}]$. For example, we can take $x=1, y=5$, and note that $1+5\sqrt{-5}$ factors as $(3+\sqrt{-5})(2+\sqrt{-5})$.

*An easier way to do this is to pick values of $y$, and then for each one, try values of $x$ until you pass $50$.
