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 \[ l(p)= O((\log p)^2\log\log p). \]


11B65 Binomial coefficients; factorials; \(q\)-identities
11P05 Waring’s problem and variants
11L40 Estimates on character sums
Full Text: DOI


[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 · doi:10.1090/S0002-9947-04-03612-8
[4] Guy, Unsolved problems in number theory (1994) · Zbl 0805.11001 · doi:10.1007/978-1-4899-3585-4
[5] DOI: 10.4064/cm96-2-4 · Zbl 1042.11002 · doi:10.4064/cm96-2-4
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.