$\displaystyle 3^k \not\equiv -1 $ mod $\displaystyle 2^e $ for e > 2, k > 0. Is this true? I have tried to prove it by expanding (1 + 2)^k. [Notation: (n; m) := n! / (m! (n - m)!)] E.g., for e = 3 I get: (1+2)^k + 1 = 2 + (k; 1) 2 + (k; 2) 2^2 + (k; e) 2^e + ... So, here it's enough to prove that 2^3 does not divide 2 + (k; 1) 2 + (k; 2) 2^2. The validity for general e seems very hard to prove.