This answer would also fit here: mathoverflow.net/questions/74214/…

Yet again, I'm happy that people managed to get these nice answers in, despite the haters wanting to close the question.

Could you please explain the line "This means that $(n^2 - 1).\alpha_A = 0$"? I don't understand exactly what that is true.

Seeing a computer proof of any facts about the sum of digits of this number is not satisfying or enlightening. Let's stop the complaining about the question and approach this problem like mathematicians, not like engineers with access to WolframAlpha.

Franz Lemmermeyer's nice articles Higher Descent on Pell Conics I, II, III may be of interest.

@AkhilMathew Do you happen to know where I can find that article nowadays? The link is dead.

Thanks! The phrase "rank one update" was very useful to know. Among other things it lead me to en.wikipedia.org/wiki/Matrix_determinant_lemma