A note on some results of Hua in short intervals |
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Authors: | Hengcai Tang |
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Institution: | 1.School of Mathematics and Information Sciences,Henan University,Henan,PR China |
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Abstract: | In this paper, we improve the previous results of the authors G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as $ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $ where U = N 1/2?η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2 k ?1, k ? 2. |
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