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Are there any odd positive numbers that satisfy the equation:

$a^2 - b^3 = 4$ ?

I am certain that there are none but can't prove it. How would you prove that?

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Start by rewriting it as $a^2-4=b^3$, and do what comes naturally.

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    @mgamer: If a prime power divides the product xy and x and y are co-prime. It has to divide either x or y. So if xy are co-prime and xy is a perfect cube _then_ (which is what you have to show. Not _if_ as you wrote, but that is true too) each x and y have to be perfect cubes.2010-10-19