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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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