# Are there composite numbers matching the conditions?

Conditions:

n such that $\ Ord_n(2) \mid n-1$ and $\ Ord_n(2) - 1 = 2^x,n \in >2\mathbb{N}+1,\ x \in \mathbb{Z}_{\geq 0}$.

I check up to 1e7 :

3,7,31,73,6151,57457,131071,599479,1572871,2252951,5242921


Numbers matching the condition are all primes.Are there composite numbers matching the conditions?

Yes: $n = 1227133513 = 23 \cdot 89 \cdot 599479$ appears to be the smallest, having $\text{ord}_n(2) = 33$. However, there seems to be no further examples up to $10^{20}$. It's not surprising that they are very rare, since pseudoprimes are already much rarer than the primes.

• Thanks for your check.Up to $\ 2^{64}\$ they say there are $\ 118968378\$ pseudoprimes,see: janfeitsma.nl/math/psp2/statistics May 29, 2014 at 7:03