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Quadratic and Divisibility Puzzle (Posted on 2015-08-09) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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a suggestive series | Comment 1 of 2
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. 

  Posted by xdog on 2015-08-09 16:46:19
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