Can the order of $2$ $mod {}$ an odd number that is a factor of $p-1$ be a multiple of the order of $2$ $mod$ $p$?

Question

Is there some odd prime $p$ and odd number $q$ such that $q|p-1$ and $ord_{p}(2)|ord_{q}(2)$? I'm fairly sure there isn't, so I'm mostly looking for a proof of that, although a counterexample would also be greatly appreciated.

Answer 1

There are examples : One of it is $$p=251$$ $$q=125$$ With the following PARI/GP - program you can search for more examples :

? forprime(p=3,10^4,fordiv(p-1,q,if(Mod(q,2)<>0,if(Mod(znorder(Mod(2,q)),znorder
(Mod(2,p)))==0,print([p,q])))))
[251, 125]
[1459, 729]
[5419, 301]
[5419, 387]
[5419, 903]
[5419, 2709]
?

By replacing "10^4" by a larger value, you can find even more examples.