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?


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

  • 1
    $\begingroup$ So I have then $(a-2)(a+2) = b^3$. I still don't see what's next... $\endgroup$ – mgamer Oct 17 '10 at 15:39
  • $\begingroup$ @mgamer: What is the gcd of $a-2$ and $a+2$? $\endgroup$ – Aryabhata Oct 18 '10 at 13:31
  • $\begingroup$ The GCD of $a+2$ and $a-2$ is? Now each of them has to be a cube individually. $\endgroup$ – Ross Millikan Oct 18 '10 at 13:32
  • $\begingroup$ Can we conclude anything about gcd of $a-2$ and $a+2$? How come? $\endgroup$ – mgamer Oct 18 '10 at 13:56
  • $\begingroup$ Well, the gcd will also divide their difference. $\endgroup$ – Derek Jennings Oct 18 '10 at 14:07

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