Garaev, Moubariz Z.; Luca, Florian; Shparlinski, Igor E. Waring problem with factorials. (English) Zbl 1076.11013 Bull. Aust. Math. Soc. 71, No. 2, 259-264 (2005). 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 \[ l(p)= O((\log p)^2\log\log p). \] Reviewer: N. Robbins (San Francisco) Cited in 2 Documents MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 11P05 Waring’s problem and variants 11L40 Estimates on character sums Keywords:Waring problem; factorials PDF BibTeX XML Cite \textit{M. Z. Garaev} et al., Bull. Aust. Math. Soc. 71, No. 2, 259--264 (2005; Zbl 1076.11013) Full Text: DOI OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.