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No integer solutions (Posted on 2012-12-10) Difficulty: 2 of 5
Prove that the equation 15x2-7y2=9 has no integer solutions.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (2 votes)

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Solution Not so fast! (spoiler) | Comment 2 of 4 |
xdog has proved that x and y cannot both be multiples of 3, but there are two additional steps needed to prove that they cannot be any other integral values:

Case 2) If y is divisible by 3 and x is not, then
                  15x^2 = 7y^2 + 9
            the right hand side is divisible by 9, but the left hand is not,               so there are no integral solutions.

Case 3) If y is not divisible by 3 then, y^2 (mod 3) = 1.  
                15x^2 = 7y^2 + 9, when reduced to mod 3,
            becomes 0 = 1, so there are no solutions

  Posted by Steve Herman on 2012-12-10 17:07:15
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