Odd values of the Ramanujan tau function

We prove a number of results regarding odd values the Ramanujan $$\tau $$ -function. For example, we existence an effectively computable positive constant $$\kappa such that if (n)$$ is and $$n \ge 25$$ then either $$\begin{aligned} P(\tau (n)) \; > \kappa \cdot \frac{\log \log {n}}{\log {n}} \end{aligned}$$ or there exists prime $$p \mid n$$ with (p)=0$$ . Here P(m) denotes largest factor m. also solve equation (n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$$ equations q^b$$ where $$3\le q < 100$$ exponents are arbitrary nonnegative integers. make use variety methods, including Primitive Divisor Theorem Bilu, Hanrot Voutier, bounds for solutions to Thue–Mahler due Bugeaud Győry, modular approach via Galois representations Frey–Hellegouarch elliptic curves.