Variations of Lehmers Conjecture for Ramanujans tau-function

We consider natural variants of Lehmers unresolved conjecture that Ramanujans tau-function never vanishes. Namely, for $n>1$ we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 691}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujans well-known congruences for $tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.