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Positive Integer Pair and Division Puzzle (Posted on 2015-10-03) Difficulty: 3 of 5
Find all pairs (x,y) of positive integers such that x divides y^2, y divides x^2 and: x+1 divides y^2+1.

No Solution Yet Submitted by K Sengupta    
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partial solution Comment 1 of 1
Any pair of the form (y^2,y) works since
y^2 divides y^2,
y divides y^4 and
y^2+1 divides y^2+1

To prove there are no other pairs you'd need to show that when x+1 divides y^2+1 it is never true that x and y have the same prime factors except when x=y^2
For example, when y=7, y^2+1=50, x+1 could be 1,2,5,10,25,50 so x could be 0,1,4,9,24,49 but none of these shares prime factors with 7 except for 49.
This seems to always work but I haven't proved it.

  Posted by Jer on 2015-10-05 14:02:47
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