Diagonal diophantine equations with small prime variables

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:440

Article

Yang, Li¹  Hu, Liqun¹ 

[1] Nanchang Univ, Dept Math, Nanchang 330031, Jiangxi, Peoples R China

关键词: Small prime;    Waring-Goldbach problem;    Circle method;   

DOI  :  10.1016/j.jmaa.2016.03.025

来源: Elsevier

【 摘 要 】

Let (a(i), a(j)) = 1, 1 <= i < j <= s and s = 2(k) + 1, where a(1), ... , a(s), s and k >= 4 are nonzero integers. In this paper, we show that if the diagonal diophantine equation a(1)p(1)(k) + ... + a(s)p(s)(k) = n is satisfying some necessary conditions, then we have the following results: For any epsilon > 0, we have (i) if a(1), .. , a(s) are not all of the same sign, then the above equation has solutions in primes p(j) satisfying p(j) << vertical bar n vertical bar(1/k) + A(3.2k-1) + epsilon, (ii) if a(1), ... , a(s) are all positive, then the above equation is solvable in prime p(j) whenever n >> A3k.2(k-1)+1+epsilon. This result is the general case of the Diophantine equations with Small Prime Variables. (C) 2016 Elsevier Inc. All rights reserved.

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