In this talk, after a brief introduction, I am planing to show how arithmetic properties of the Ramanujan tau function $\tau(n)$ can be combined with the Waring-Goldbach problem to prove that the set of the values $\tau(n)$ forms a finite additive basis for the set of integers. Moreover, for every integer $|N|\ge 2$, the Diophantine equation $$ \tau(n_1)+ \ldots + \tau(n_{148000})=N, $$ has a solution in positive integers $n_1, \ldots, n_{148000},$ satisfying the condition $$ \max_{1\le i \le 148000}n_i \ll |N|^{2/11}e^{-c \log |N|/\log \log|N|}, $$ for some absolute constant $c >0.$ In view of a result of Deligne, this result reflects the best possible order of the integers $n_i,$ apart from the value of the constant $c.$