Heilbronn ’ S Conjecture on Waring ’ S Number ( Mod P )
نویسنده
چکیده
Let p be a prime k|p−1, t = (p−1)/k and γ(k, p) be the minimal value of s such that every number is a sum of s k-th powers (mod p). We prove Heilbronn’s conjecture that γ(k, p) k1/2 for t > 2. More generally we show that for any positive integer q, γ(k, p) ≤ C(q)k1/q for φ(t) ≥ q. A comparable lower bound is also given. We also establish exact values for γ(k, p) when φ(t) = 2. For instance, when t = 3, γ(k, p) = a+ b− 1 where a > b > 0 are the unique integers with a2 + b2 + ab = p, and when t = 4, γ(k, p) = a− 1 where a > b > 0 are the unique integers with a2 + b2 = p.
منابع مشابه
Sum - Product Estimates Applied to Waring ’ S Problem Mod
Let γ(k, p) denote Waring’s number (mod p) and δ(k, p) denote the ± Waring’s number (mod p). We use sum-product estimates for |nA| and |nA − nA|, following the method of Glibichuk and Konyagin, to estimate γ(k, p) and δ(k, p). In particular, we obtain explicit numerical constants in the Heilbronn upper bounds: γ(k, p) ≤ 83 k, δ(k, p) ≤ 20 k for any positive k not divisible by (p− 1)/2.
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