On values of Ramanujan’s tau function involving two prime factors

Abstract

Lehmer’s conjecture that $\tau (n)$ never vanishes remains open. In the spirit of this conjecture, it is natrual to consider for a given integer $\alpha$, whether there exists $n\in \mathbb {Z}_+$ such that $\tau (n)=\alpha$. Several recent papers exclude many odd $\alpha$ involving one prime factor or small values of the $\tau$-function using the theory of congruences of $\tau (n)$, Lucas sequences, integer points of elliptic curves, and the Thue equation. In this paper, we generalize the results and methods to infinitely many odd $\alpha$ involving two prime factors, showing that $\tau (n)\ne \pm p_1{p_2}^b$ for arbitrary $b\in \mathbb {Z}$, where $3\le p_1, p_2\le 31$ are primes such that $p_1p_2\le 247$ or $p_1=p_2$. Moreover, together with the previous results, we get $\tau (n)\ne 3^{b_1}5^{b_2}7^{b_3}11^{b_4}13^{b}$ for $b\in \mathbb {Z}$, and $b_i\in \{0, 1\}$, $i=1, 2, 3, 4$. Besides, we show that $|\tau (n)|> |\tau (3)|$ for all odd $\tau (n), n\ne 1$.

Appendix

See Tables 5 and 6.

Table 5 All nonempty sets $H_j(f)$

Table 6 The values of m, a and the congruences for $\tau (p^a)\ne m$