Waring-Goldbach problem for fourth powers with almost equal variables. (English) Zbl 1354.11066
Summary: We consider the expression of a positive integer n by sums of fourth powers of almost equal primes, that is, \(n = p_{1}^{4} + p_{2}^{4} + \cdots + p_{s}^{4}\) 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 \(\theta_{17} = 1/196-\varepsilon\). Moreover, we prove that when \(9\leq s\leq 16\), almost all integers n can be expressed this way with \(\theta_{s} = (2s-15)/(12(s + 16))-\varepsilon\) for \(s = 9, 10\) and \((8s-69)/(4(88s-717))-\varepsilon\) for \(11\leq s\leq 16\).
MSC:
| 11P32 | Goldbach-type theorems; other additive questions involving primes |
| 11P05 | Waring’s problem and variants |
| 11P55 | Applications of the Hardy-Littlewood method |
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