Lehmer conjectured that Ramanujan's tau function never vanishes. As a variation of this conjecture, it is proved that \begin{ equation*} \tau(n)\neq \pm \ell, \pm 2\ell, \pm 2\ell^2, \end{ equation*} where $\ell<100$ is an odd prime, by Balakrishnan, Ono, Craig, Tsai and many people. We have proved that \begin{ equation*} \tau(n)\neq \pm \ell, \pm 2\ell, \pm 4\ell, \pm 8\ell \end{ equation*} for any $n\geq 1$ except 14 cases, where $\ell<1000$ is an odd prime.
Categories: math.NT