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Heilbronn's conjecture on Waring's number (mod p)

✍ Scribed by James Arthur Cipra; Todd Cochrane; Christopher Pinner

Book ID
104024843
Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
133 KB
Volume
125
Category
Article
ISSN
0022-314X

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✦ Synopsis

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 kth powers (mod p). We prove Heilbronn's conjecture that γ (k, p) k 1/2 for t > 2. More generally we show that for any positive integer q, γ (k, p) C(q)k 1/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 a 2 + b 2 + ab = p, and when t = 4, γ (k, p) = a -1 where a > b > 0 are the unique integers with a 2 + b 2 = p.

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