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Positive Pair Poser (Posted on 2015-05-09)

Determine all possible pairs (x,y) of positive integers with gcd(4x+1, 4y-1) = 1 such that x+y divides 16xy+1

No Solution Yet Submitted by K Sengupta

Rating: 4.0000 (1 votes)

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Two unhelpful(?) observations

| Comment 1 of 8

Observation:

16xy+1 = 16xy + 16y^2 - 16y^2 + 1

= 16y(x+y) + (4y+1)(4y-1)

So, if x+y divides 16xy+1, it also divides (4y+1)(4y-1)

Similarly, it also divides it also divides (4x+1)(4x-1)

Not clear yet how that helps

/******************************/

Different observation.

16xy+1 is odd, so x+y must be odd.

so either x is even and y is odd, or x is odd and y is even

Posted by Steve Herman on 2015-05-09 14:03:08

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