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Ratio Sum Equals Positive Integer (Posted on 2015-08-30) Difficulty: 3 of 5
Does there exist an infinite number of positive integer pairs (x, y) such that:
(x+1)/y + (y+1)/x is a positive integer?

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No Solution Yet Submitted by K Sengupta    
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Solution Answer, maybe full solution. | Comment 1 of 3
First if you check using f(x,y) -> (x+1)/y + (y+1)/x
you find that
f(y,y(y+1)/x) comes out to the same thing (algebra not shown.)
so if you find one solution you get infinitely many.

I found two families based on
f(2,3)=3  and f(1,2)=4
but no others up to 10.

The first few terms that yield 3 are
(2,2)
(2,3)
(3,6)
(6,14)
(14,35)
(35,90)

The first few terms that yield 4 are
(1,1)
(1,2)
(2,6)
(6,21)
(21,77)
(77,286)

In trying to find other yields I checked up to 10 using 100 as the smaller number.  There are sum interesting near misses such as
f(88/870) = 4399/440 (close to 10) which may warrant further investigation.

Edited because I forgot the first pair of each list.

Edited on August 30, 2015, 8:51 pm
  Posted by Jer on 2015-08-30 16:47:23

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