Playing with Powers of 2

Let $P\left( x \right)$ be the sum of the digits of $x$ . We define $T\left( x\right)$ as applying $P\left( x \right)$ to the number $x$ , and then to the result (repeatedly), until the outcome is a single digit. For example, $T\left( 249 \right) :\quad P(249)=15,\quad P(15)=6$ so $T\left( 249 \right) =6$ .

Let $T({ 2 }^{ 2016 })=N$ . Find the least possible value for $x>1$ that satisfy ${ 2 }^{ P(x) }\equiv N \pmod x$ .