# Thread: Prove prime test conjecture.

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

2. Originally Posted by RLBrown
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.