Waring problem with factorials.(English)Zbl 1076.11013

If $$p$$ is an odd prime, let $$l(p)$$ be the least integer $$l\geq 1$$ such that for all $$\lambda$$ such that $$0\leq\lambda\leq p-1$$, the congruence $(n_1)!+ (n_2)!+\cdots+ (n_l)!\equiv\lambda\pmod p$ has a solution in positive integers $$n_1,n_2,\dots, n_l$$. The authors prove that $l(p)= O((\log p)^2\log\log p).$

MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 11P05 Waring’s problem and variants 11L40 Estimates on character sums

Keywords:

Waring problem; factorials
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References:

 [1] Vinogradov, Elements of number theory (1954) [2] Stewart, Publ. Math. Debrecen 65 pp 461– (2004) [3] DOI: 10.1090/S0002-9947-04-03612-8 · Zbl 1060.11046 [4] Guy, Unsolved problems in number theory (1994) · Zbl 0805.11001 [5] DOI: 10.4064/cm96-2-4 · Zbl 1042.11002
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