## Products of factorials modulo $$p$$.(English)Zbl 1042.11002

Let $$P_{s,t}(p)=\{x_1!\dots x_t!\pmod{p} \mid x_i\geq1 \text{ for } i=1,\dots,t \text{ and } \sum_{i=1}^t x_i=s \}$$. Two main results of the paper say: (1) Let $$\varepsilon>0$$ be arbitrary. There exists a computable positive constant $$p_0(\varepsilon)$$ such that whenever $$p>p_0(\varepsilon)$$, then $$P_{s,t}(p)$$ covers all the non-zero residue classes modulo $$p$$ for all $$t$$ and $$s$$ such that $$t>p^\varepsilon$$ and $$s-t>p^{1/2+\varepsilon}$$; (2) If $$p\neq 5$$ is a prime, then the set $$\bigcup_{t\geq2} P_{s,t}(p)$$ covers all the non-zero residue classes modulo $$p$$.

### MSC:

 11A07 Congruences; primitive roots; residue systems 11N69 Distribution of integers in special residue classes 11B65 Binomial coefficients; factorials; $$q$$-identities
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