Has $\ 2^{n-1}\equiv 2^{41}+1\mod n\ $ a solution?

Has $$2^{n-1}\equiv 2^{41}+1\mod n$$ a solution with a positive integer $\ n>1\ $ ?

Motivation : The equation $$2^{n-1}\equiv k\mod n$$ has always a solution, if $\ k-1\ $ has an odd prime factor (this odd prime factor is then a solution) and for $\ k=2^m+1\ $ , I know a solution for $$m=1,2,3,\cdots,40$$ Hence, this is the smallest number for which I know no solution. Upto $\ n=10^9\ $, there is no solution.

Solution 1:

Yes, $n=24189255799819$ is a solution (may be not the smallest one). Tried to search for $n=pq$ with $p,q$ prime and $p$ small, by factoring $2^p-2k$, and got it already at $p=11$.