There exist no Ramanujan congruences $\mod691^2$

A. A. Panchishkin

Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 255-263

Résumé

Let $\tau(n)$ be Ramanujan's function, $$ x\prod_{m=1}^\infty(1-x^m)^{24}=\sum_{n=1}^\infty\tau(n)x^n. $$ In this paper it is shown that the Ramanujan congruence $\tau(n)\equiv\sum_{d/n}d^{11}\bmod691$ cannot be improved $\bmod691^2$. The following result is proved: for arbitrary $r$, $s\bmod691$ the set of primes such that $p\equiv r\bmod691$, $\tau(p)\equiv p^{11}+1+691\cdot s\bmod691^2$ has positive density.