Geodesic
There exist no Ramanujan congruences $\mod691^2$
Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 255-263 
A. A. Panchishkin. There exist no Ramanujan congruences $\mod691^2$. Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 255-263. http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a7/
@article{MZM_1975_17_2_a7,
     author = {A. A. Panchishkin},
     title = {There exist no {Ramanujan} congruences $\mod691^2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {255--263},
     year = {1975},
     volume = {17},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.