A Mersenne prime is a prime number of the form $M_p = 2^p -1$, for $p$ a prime number. We have the sequence $$s_i = \begin{cases} 4, \text{ if } i = 0;\\ s_{i-1}^2 - 2 \text{ otherwise. }\end{cases}$$ The Lucas-Lehmer primality test says that $M_p$ is prime if and only if $s_{p-2} \equiv 0 \pmod{M_p}$. I was wondering if anyone could provide any justification for where this sequence comes from?
Wikipedia has a relatively simple proof of this fact proven by J.W. Bruce and Öystein J. Rödseth, but it gives no justification of the choice of sequence. I've looked at D.H. Lehmer's paper "An Extended Theory of Lucas' Functions", which contains the original proof of this statement. Lehmer gives a family of sequences $U_n = (a^n - b^n)/(a-b)$, where $a$ and $b$ are the roots of $x^2 - \sqrt{R}x + Q=0$. In his proof for the test of Mersenne numbers (Theorem $5.4$ in the paper), he chooses the sequence given by $R = 2, Q = -1$. Given the sequence, it is clear to me how we got the sequence as given above. If anybody would be able to provide some insight as to why either the sequence $s_i$ was chosen, or the choice of polynomial $x^2 - \sqrt{2}x -1$ is chosen, I would be greatly appreciative.
