projecteuler.net

Powers of $1+\sqrt 7$

 Published on Sunday, 21st March 2021, 01:00 am; Solved by 257

Problem 752

When $(1+\sqrt 7)$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt 7)$.
We write $(1+\sqrt 7)^n = \alpha(n) + \beta(n)\sqrt 7$.

For a given number $x$ we define $g(x)$ to be the smallest positive integer $n$ such that: $$\begin{align} \alpha(n) &\equiv 1 \pmod x\qquad \text{and }\\ \beta(n) &\equiv 0 \pmod x\end{align} $$ and $g(x) = 0$ if there is no such value of $n$. For example, $g(3) = 0$, $g(5) = 12$.

Further define $$ G(N) = \sum_{x=2}^{\strut N} g(x)$$ You are given $G(10^2) = 28891$ and $G(10^3) = 13131583$.

Find $G(10^6)$.