# Range of $2^n$ mod $n$

Is there any integer $$n\ge 2$$ such that $$2^n\equiv 3 \bmod n$$? I understand that $$n$$ must be an odd non-prime. I checked up to a million with no success (but $$2^n\equiv 5 \bmod n$$ and $$2^n\equiv 7 \bmod n$$ have solutions).

The smallest such $n$ is $n=4700063497$. A few others are known. J. Crump found $n=8365386194032363$ in 2000. Max Alekseyev found $n=3468371109448915$. Joe Crump found $n=10991007971508067$. Some information on these is at http://www.cs.ucla.edu/~klinger/newno.html
• And using OEIS, one can find least solutions for $2^n=2k+1\pmod n$ for $k=1,\ldots,33$: oeis.org/A124977 – Yoav Kallus Jun 28 '13 at 2:55