How to prove that all odd powers of two add one are multiples of three 
For example
  \begin{align} 
2^5 + 1 &= 33\\
2^{11} + 1 &= 2049\ \text{(dividing by $3$ gives $683$)}
\end{align}

I know that $2^{61}- 1$ is a prime number, but how do I prove that $2^{61}+1$ is a multiple of three?

If $k$ is odd, then $2^k$ is $2*2^{k-1}$. If $n$ is not a multiple of 3 then $n^2-1$ is since $n^2-1 = (n-1)(n+1)$ and at least one of them is a multiple of 3. This means that $2*2^{k-1} - 2$ is a multiple of $3$ as is $2*2^{k-1} + 1$.
Hope I helped.
