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Expression Divides Expression (Posted on 2015-09-12) Difficulty: 3 of 5
Does there exist infinitely many pairs of distinct positive integers (A,B) such that A2 + B3 is divisible by A3 + B2?

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No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
Let B=A*C.  Then (A^2+B^3)/(A^3+B^2) = (1+A*C^3)/(A+C^2).

Then some manipulation:
(1+A*C^3)/(A+C^2)
= (A*C^3+C^5-C^5+1)/(A+C^2)
= C^3 + (1-C^5)/(A+C^2)
= C^3 + (1-C)*(C^4+C^3+C^2+C+1)/(A+C^2)

Setting A=C^4+C^3+C+1 will cancel out the denominator with the large factor in the numerator, which reduces the expression to C^3-C+1.

Then for any positive integer C, (A, B)=(C^4+C^3+C+1, C^5+C^4+C^2+C) will make (A^2+B^3)/(A^3+B^2) = C^3-C+1.  This proves there is an infinite number of integer solutions.

  Posted by Brian Smith on 2017-06-24 11:45:22
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