perplexus.info :: Just Math : No integer solutions

No integer solutions (Posted on 2012-12-10)

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/spoiler

| Comment 1 of 4

Set x=3a and y=3b giving

15a^2 - 7b^2 = 1.

Write this as 15a^2 - 6b^2 = b^2 + 1.

LHS is evenly divisible by 3 so RHS must be too.

But b = 0,1,-1 mod 3 and b^2 + 1 = 1,2 mod 3, not 0.

Consequently, there are no integer solutions.

Posted by xdog on 2012-12-10 16:11:43

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