Abstract
We consider the expression of a positive integer n by sums of fourth powers of almost equal primes, that is, n = p 41 + p 42 + ⋯ + p 4 s with \( \left|{p}_i-{\left(N/s\right)}^{1/4}\right|\leqslant {\left(N/s\right)}^{1/4-{\theta}_s} \). We establish that for every sufficiently large integer N satisfying necessary local conditions, this equation holds with s = 17 and θ17 = 1/196 − ε. Moreover, we prove that when 9 ⩽ s ⩽ 16, almost all integers n can be expressed this way with θs = (2s − 15)/(12(s + 16)) − ε for s = 9, 10 and (8s − 69)/(4(88s − 717)) − ε for 11 ⩽ s ⩽ 16.
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This work is supported by Natural Science Foundation of Shandong Province (grant No. ZR2015AM016), Natural Science Foundation of China (grant No. 11401344), and Natural Science Foundation of China (grant No. 11501324).
An erratum to this article is available at http://dx.doi.org/10.1007/s10986-017-9349-0.
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Yao, Y., Zhang, M. Waring–Goldbach problem for fourth powers with almost equal variables. Lith Math J 56, 572–581 (2016). https://doi.org/10.1007/s10986-016-9337-9
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DOI: https://doi.org/10.1007/s10986-016-9337-9