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We show that for infinitely many prime numbers p there are at least \log\log p / \log\log\log p distinct residue classes modulo p that are not congruent to n! for any integer n.
Let p(n) denote the number of partitions of n . Ramanujan's partition congruences are p(5n + 4) , p(7n + 5) and p(11n + 6) = mod 5, 7, and 11, respectively. These have been proved in number of ways. ...