Is there any integer N such that 2^N=3 mod N? I understand that N must be an odd non-prime. I checked up to a million with no success (but, FYI, 2^N=5 and 2^N=7 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