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Cubic Expression and Perfect Cube (Posted on 2015-10-04)

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.

See The Solution Submitted by K Sengupta

Rating: 3.6667 (3 votes)

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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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