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Expression Quotient Puzzle (Posted on 2015-06-16) Difficulty: 3 of 5
Find all pairs (x,y) of positive integers x and y such that:
x + y + 1 divides 2xy and:
x + y − 1 divides x2 + y2 − 1

No Solution Yet Submitted by K Sengupta    
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Some, maybe all | Comment 1 of 3
By symmetry if (x,y) works then (y,x) works.

If y=x+1 then
x+(x+1)+1 = 2(x+1) divides 2x(x+1) and
x+(x+1)-1 = 2x divides x2 + (x+1)2 − 1 = 2x(x+1)

so all of (x,x+1) and (x,x-1) works.

There appear to be other sporadic cases to each of the two divisions, but I haven't found any in common yet.

  Posted by Jer on 2015-06-16 09:30:04
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