Does there exist an infinite number of pairs (A,B) of positive integers such that:
 A divides B^{2} + 1 and:
 B divides A^{2} + 1?
Give valid reasons for your answer.
Take A<B. Since we have to start somewhere, set A=1 and B=2. As it happens this pair works.
Then for subsequent pairs, set A(n)=B(n1) and assign B(n) to be the smallest factor of A(n)^2 > A(n).
We get this series:
n A(n) B(n)
1 1 2
2 2 5
3 5 13
4 13 89
5 89 233
? ??? ???
in which each term is an oddnumbered term of the Fibonacci series.

Posted by xdog
on 20150809 16:46:19 