Alon Levy points that equation I’m interested in,
is called Ramanujan equation. (You must know who Ramanujan was.) He also gives a full solution (which I haven’t verified yet) and a link to alternative solution. I’ve checked in Alan Baker’s Concise introduction to the theory of numbers that indeed it was one of Ramanujan’s many conjectures that above equation has solutions only for n=3,4,5,7 and 15. It was proved by T. Nagell in 1948. (Precise references are given here.)
However, I still have an impression that something funny is going on.
Namely, it’s rather straightforward to verify that above has a solution for given n iff sequence bk defined by
b1=b2=1, bk+2 = bk+1 – 2bk
takes value +-1 for k=n-2. This sequence can be also, by a standard method, put in a closed form:
bk=1/√7 ∙ (ωn – ψn),
where ω and ψ are roots of X2+2X-1.
So, to be happy we need to prove that |bk| > 1 for k>13. ‘Till recently I thought that it’s an arithmetical problem, but yesterday I ordered my computer to give me first 5000 bk‘s. This is what he gave me, on horizontal axis is k and on vertical is ln|bk|.
Evidently, that bk is rarely equal to +-1 is not a matter of hard arithmetics of Z[ω], but rather of the fact that |bk| are roughly equal to const∙ek, which rarely is a number around +-1! Still, I have a quite explicit and quite simple formula for bk, a strong numerical evidence that it goes to infinity but can’t prove nothing about this sequence! Moreover, even Ramanujan apparently didn’t know much about it. This is funny :-). If you have any guts, you should try to prove that bk goes to infinity by some simple method ;-).
And general question: is it true that any Fibonnaci-like sequence is either bounded or its norm goes exponentially to infinity?
Maxim Kontsevitch was giving a lecture today. I hope to write what I understood tomorrow.