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I am struggling with this strange problem.

I have to find pairs of positive integers that satisfy two specific conditions.

1. you can’t go from one pair to another by subtracting/adding the same number to both members of the pair.

2. You can’t go from one pair to another by subtracting/adding a number from one of the numbers in the pair.

I have so far found the following pairs using a graph paper. Note that the order of the numbers in a pair is not important.

(0,0) (1,2) (3, 5) (4, 7) (6, 10) (8, 13)

I want to able to find say 100th pair. How should I go about doing this?

Also, what branch of mathematics handles this kind of problems?

Thanks.