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How do I prove $n$ is a Carmichael number?

I am trying to verify the fact that $1729$ is a Carmichael number. However, a number $n$ is a carmichael number if and only if $a^{n-1} \equiv 1 \pmod n$ for all $a$ that are relatively prime to $n$. Is there any more efficient way of showing that it is a carmichael number short of testing all of the numbers that are relatively prime to $1729$?

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Shanks Baroda
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  • Not quite, that post qualifies the nature of a general carmichael number, while this seeks to show that the characteristics presented in the other post hold in the case of 1729. – Shanks Baroda Jan 23 '13 at 04:19
  • I agree they are not identical, but doesn't the general question subsume the specific one? It immediately shows that there is a more efficient way to verify Carmichaelness, which was your question. – Erick Wong Jan 23 '13 at 04:31

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