If we say $n=p_1^{\alpha_1}\times p_2^{\alpha2}\times \cdots \times p_k^{\alpha_k}$, where $p_i$ are prime numbers, $\alpha_i$ are natural numbers, can or can we not say that:
Choose a $p_i$ such that it minimizes the quantity of $v_2(p_i-1)$.
1) Write $p_i=1+2^{\beta_i}\gamma_i$, where $\gamma_i$ is an odd integer, then $n\equiv 1 (\mbox{mod }2^{\beta_i})$ (I actually copied this out from a book. Just curious why is it true.)
Indeed, if the above is true, $n-1=2^{\beta_i}t$, for some integer $t$.
2) Is this true and why: $2^{2^{\beta_i}t}\equiv -1 (\mbox{mod }p_i)$