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.
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(n-1) 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 odd-numbered term of the Fibonacci series.
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Posted by xdog
on 2015-08-09 16:46:19 |
a suggestive series
