MHB Is the Binomial Coefficient Test a Reliable Prime Indicator?

AI Thread Summary
The discussion centers on a conjecture regarding the binomial coefficient test as a prime indicator for odd integers greater than 2. It states that an odd integer n is prime if and only if the binomial coefficient $\left( \begin{array}{c} n-1 \\ \frac{n-1}{2} \end{array} \right) \equiv \pm 1 \mod n$. While this condition holds true for prime numbers, it also applies to certain composite numbers like 5907, 1194649, and 12327121, indicating the conjecture is not universally valid. The existence of other non-prime odd numbers that meet this condition remains unknown. Further exploration into "Catalan pseudoprimes" may provide additional insights.
RLBrown
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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
 
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RLBrown said:
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 $${n-1 \choose \frac{n-1}2} \equiv \pm1\!\!\! \pmod n$$ 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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