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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
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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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