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    #1
    I was examining the AKS and discovered this conjecture.

    Please prove the following true or false.
    Let n be an odd integer >2

    then n is prime IFF
    $\left(
    \begin{array}{c}
    n-1 \\
    \frac{n-1}{2} \\
    \end{array}
    \right)

    \text{ $\equiv $ }
    \pm 1$ mod n
    Last edited by RLBrown; July 24th, 2016 at 05:43.

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    #2
    Quote Originally Posted by RLBrown View Post
    I was examining the AKS and discovered this conjecture.

    Please prove the following true or false.
    Let n be an odd integer >2

    then n is prime IFF
    $\left(
    \begin{array}{c}
    n-1 \\
    \frac{n-1}{2} \\
    \end{array}
    \right)

    \text{ $\equiv $ }
    \pm 1$ mod n
    A quick internet search indicates that this result is false, but only just! In fact, the condition $ \displaystyle {n-1 \choose \frac{n-1}2} \equiv \pm1\!\!\! \pmod n$ holds whenever $n$ is prime. However, it also holds for the numbers $5907$, $1194649$ and $12327121$, which are not prime. It is not known whether there are any other non-prime odd numbers that satisfy the condition.

    For more information, search for "Catalan pseudoprimes".

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