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Cubic Expression and Perfect Cube (Posted on 2015-10-04) Difficulty: 3 of 5
Each of x and y is a positive integer.
Can each of x^3+y and y^3+x be a perfect cube?.
Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution q.e.d. (spoiler) | Comment 1 of 2
No, they cannot both be a perfect cube.

Without loss of generality, assume that y >= x.

Then y^3 + x cannot be a perfect cube, because it is greater than y^3 but less than (y+1)^3.

To see this, note that (y+1)^3 is y^3 + 3y^2 + 3y + 1, which is obviously greater than y^3 + x whenever y >= x.

q.e.d.

  Posted by Steve Herman on 2015-10-04 20:22:25
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