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Perfect square? (Posted on 2004-07-23) Difficulty: 4 of 5
Find all integer solutions of y = x - 432.

See The Solution Submitted by Nick Hobson    
Rating: 3.0000 (6 votes)

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Solution re: Answer Comment 12 of 12 |
(In reply to Answer by K Sengupta)

y^2 = x^3 - 432
or, y^2 + 432 = x^3
or, (y/6 + 6)^3 = (y/6 -6)^3 + x^3

Since y is an integer, (y/6 +/- 6) is always a rational number irrespective of whether y is divisible by 6 or not.

But, by Fermat's last Theorem, the sum of cubes of two nonzero rational numbers cannot be equal to the cube of a nonzero rational number.

Since each of x and (y/6 +/- 6) is a rational number, the equality in (*) is only possible, only when:
either, y/6 -6 = 0, giving y = 36, so that x= 12

or, y/6 + 6 = 0, giving y = -36, so that x= 12

or, x = 0, giving: y^2 = - 432 from the original equation, so that y is imaginary, which is a contradiction.

Consequently, (x, y) = (12, 36) and (12, -36) are the only possible solutions to the given problem. 

Edited on June 10, 2008, 6:25 am
  Posted by K Sengupta on 2008-06-10 06:19:23

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