it seems to me that there are on solutions for x>3,I have completed a computer search for x>3 and x<=1 billion.  Now I think that x=1 could be the solution depending on what is meant by the equal sign in 2^x mod x = 3.  If it means exact inequality then 2^1 mod 1=0 is not = to 3.  But if it means congruence then 3 mod 1 =0 and 2^1 mod 1 =0 and thus 2^1 mod 1= 3.