Prove that $\displaystyle \left({m\choose 0} + {m\choose 1} - {m\choose 2} - {m\choose 3}\right) + \left({m\choose 4} + {m\choose 5} - {m\choose 6} - {m\choose 7}\right) + \mbox{...} = 0$ if and only if for some positive integer $\displaystyle k,\ m = 4k - 1$