Does there exist an infinite number of pairs (A,B) of positive integers such that:
- A divides B2 + 1 and:
- B divides A2 + 1?
Give valid reasons for your answer.
(In reply to a suggestive series
First off your series contains an error: you left out 34.
Take n=3 for an example, where A(3)=F(5) and B(3)=F(7).
what your really found is F(7)^2+1=F(5)*F(9) and F(5)^2+1=F(3)*F(7)
Which is the odd-numbered Fibonacci terms as you pointed out.
What's left is to show F(n)^2+1 = F(n-2)*F(n+2) for the odd terms.
I'll leave out the tricky math but these are the results.
As you can see when n is odd,
F(n)^2+1=[(phi)^(2n)+phi^(-2n)+2*(-1)]/5+1 = [(phi)^(2n)+(phi)^(-2n)+7]/5
but if n is even instead the result is [(phi)^(2n)+(phi)^(-2n)+3]/5
Posted by Jer
on 2015-08-09 22:06:40