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I'm studying about Pell number and Pell-Lucas number whose have Binet formula $P_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ and $Q_n=\alpha^n+\beta^n$, where $\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$, respectively.

The simple relation between two of those is $Q^2_n-8P_n^2=4(-1)^n$.

Notice that if $q$ is odd prime factor of $P_n$ and $n$ is odd, we have $Q^2_n \equiv -4 \mod q$. "Because $q$ is odd, then $q \equiv 1 \mod 4$".

I have no idea why $q \equiv 1 \mod 4$. Does it able to be $3\mod 4$?

Thank you for attention.

Reference: Pell numbers with the Lehmer property, Bernadette Faye and Florian Luca. Afr. Mat. 2017 28:291-294

Authawich Narissayaporn
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  • Let $A_n=\frac{Q_n}{2}$, it is an integer. If $n$ is odd and $q$ an odd prime $q ,|, P_n^2$ then $q, |, A_n^2 +1$. Thus $-1 \equiv A_n^2 \bmod q$ so $-1$ is a square modulo $q$. And since no square is $\equiv 2 \bmod 4$ it means $q \equiv 1 \bmod 4$. To go further you'll need the quadratic reciprocity and the multiplicative group structure of $\mathbb{Z}/q \mathbb{Z}[\sqrt{2}]$. – reuns Oct 03 '17 at 10:18
  • @reuns Thank you for making me clear. :D – Authawich Narissayaporn Oct 04 '17 at 10:41

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