There exist no Ramanujan congruences mod 6912 |
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Authors: | A. A. Panchishkin |
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Affiliation: | 1. M. V. Lomonosov Moscow State University, USSR
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Abstract: | Let τ(n) be Ramanujan's function, $$xprod _{m = 1}^infty (1 - x^m )^{24} = sumnolimits_{n = 1}^infty {tau (n)x^n .} $$ In this paper it is shown that the Ramanujan congruence τ(n)=σd/nd11 mod 691 cannot be improved mod 6912. The following result is proved: for arbitrary r, s mod 691 the set of primes such that p ≡ r mod 691,τ (p) ≡ p11+1+691 · s mod 6912 has positive density. |
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