Is there any integer $N$ 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).

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).

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).

Is there any integer $N$ 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).