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摘要翻译：
设k>2为固定整数指数，设\theta>9/10。我们证明了一个正整数N可以表示为3k次幂的非平凡和或差，使用大小至多为B的整数，用O(B^{θ}N^{1/10}）的方式，条件是N<<B^{3/13}。这一点的意义在于，我们可以严格地取θ小于1。我们还证明了O(B^{10/k}）,（以N<<B)为前提，它对大k较好。结果推广到任意固定非奇异三元的表示。然而，“非微不足道的”必须得到适当的定义。考虑奇异形式x^{k-1}y-z^k使我们建立了当p在素数上运行时p^k+c的(k-1)自由值的渐近公式，回答了Hooley提出的一个问题。
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英文标题：
《Sums and Differences of Three k-th Powers》
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作者：
D.R. Heath-Brown
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take \theta strictly less than 1. We also prove the estimate O(B^{10/k}), (subject to N << B) which is better for large k. The results extend to representations by an arbitrary fixed nonsingular ternary from. However ``non-trivial'' must then be suitably defined. Consideration of the singular form x^{k-1}y-z^k allows us to establish an asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes, answering a problem raised by Hooley.
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PDF链接：
https://arxiv.org/pdf/0806.4330
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