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

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

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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