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Rational solutions of Pell's equation

The difference equation given by initial conditions $a_0=0$, $b_0=1$ and the recursion: $a_{n+1}=A\times a_n+B\times b_n$ and $b_{n+1}=N\times B\times a_n+A\times b_n$ Gives solutions to: $(b_n)^2-N\...
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Find all primes $p$ and $q$ such that $p^2-2q^2=1.$

One more approach (just for exercising). We use the knowledge that $1=3^2-2 \cdot 2^2$ and rewrite the initial formula as $$ p^2 - 2q^2 = 3^2-2 \cdot 2^2 \tag 1$$ and then $$ p^2 - 3^2 = 2(q^2-2^2) \...
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