Marc LeBrun writes: <<
=Dan Asimov ...QUESTION: What is a formula for fp(n) in general?
Interesting question, but isn't this basically asking for a formula for the multiplicative order of the matrix [1 1] [0 1] mod n? I understand that's hard for just integers; should we expect it to be easier in matrix land? (but fp(n) sounds like a great candidate for the OEIS!)
Maybe so about the matrices, but it appears there *is* a formula for fp(n) -- which turns out (no surprise!) to already be in OEIS. The OEIS entry mentions that for a prime p, (*) fp(p) = (p-1). Can someone please explain this notation to me? Thanks. Another comment at that entry is that if n = p1^e1*...* pk^ek, for distinct primes p1,..., pk, then (**) fp(n) = LCM(fp(p1^e1),...,fp(pk^ek)). And I've just noticed strong numerical evidence that for a prime power n = p^r, we have (***) fp(p^r) = p^(r-1) * fp(p). If (***) can be proved, then putting these three equations together gives essentially a formula for all fp(n). (But what the heck does (p-1) mean?!?!?) --Dan