Fermat's divisibility conjecture for Mersenne numbers

Let $p$ be an odd prime, and define the least occurrence exponent $k$ to be the least positive integer such that $p$ divides the Mersenne number $2^k-1$, when such an integer exists. Fermat's divisibility conjecture. For every odd prime $p$, the least occurrence exponent $k$ exists and satisfies

$$k\mid(p-1).$$

Equivalently, every odd prime divides $2^{p-1}-1$, and if it first divides $2^k-1$, then $k$ is a divisor of $p-1$. This is presented as Fermat's October statement for the base $2$ and as the broader claim from which Fermat's theorem follows.