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
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.
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".