Waring’s number mod \(m\). (English) Zbl 1217.11086
Summary: Let \(p\) be an odd prime and \(\gamma (k,p^n)\) be the smallest positive integer \(s\) such that every integer is a sum of \(s\) \(k\)th powers (mod \(p^n\)). We establish \(\gamma (k,p^n)\leq [k/2]+2\) and \(\gamma (k,p^n)\ll \sqrt k\) provided that \(k\) is not divisible by \((p - 1)/2\). Next, let \(t=(p - 1)/(p - 1,k)\), and \(q\) be any positive integer. We show that if \(\phi(t)\geq q\) then \(\gamma (k,p^n)\leq c(q)k^{1/q}\) for some constant \(c(q)\). These results generalize results known for the case of prime moduli.
MSC:
| 11P05 | Waring’s problem and variants |
References:
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