Results 11 to 20 of about 496 (82)
Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem [PDF]
Open Mathematics, 2017 In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., N=p13+…+pj3$\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with |pi−(N ...
Feng Zhao
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On the Waring–Goldbach problem for seventh and higher powers [PDF]
Monatshefte Fur Mathematik, 2016 We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function $H(k)$ in the Waring-Goldbach problem. We obtain new results for all exponents $k \ge 7$, and in particular establish that for large $k$ one has \[H(k)\le (4k ...
Kumchev, Angel, Wooley, Trevor D
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On Waring–Goldbach problem with Piatetski-Shapiro primes [PDF]
Journal of Number Theory, 2018 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Yildirim Akbal
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On the Waring--Goldbach problem for eighth and higher powers [PDF]
Journal of the London Mathematical Society, 2015 Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem.
Angel V. Kumchev, D. Wooley, Trevor
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A note on the Waring–Goldbach problem
Journal of Number Theory, 2010 For a positive integer \(k\) and a prime \(p\), let \(\nu_p(k)\) denote the \(p\)-adic valuation of \(k\), i.e., \(\nu_p(k)\) is the largest integer \(\nu\) such that \(p^{\nu} \mid k\). Let \(\gamma(k, p) = \nu_p(k)+1\) if either \(p\) or \(k\) is odd, and let \(\gamma(k, 2) = \nu_2(k)+2\) if \(k\) is even. For a positive integer \(k\), we define \(K =
Jack Buttcane
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Waring-Goldbach problem in short intervals
Israel Journal of Mathematics, 2019 Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions, can be ...
Wang, Mengdi
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On the Waring–Goldbach problem with small non-integer exponent [PDF]
Acta Arithmetica, 2003 Let \(c>1\) be non-integer and denote by \(H(c)\) the least \(k\) such that the inequality \[ |p_1^c + p_2^c + \cdots + p_k^c - N|< \varepsilon \] has a solution in prime numbers \(p_1, p_2, \ldots, p_k\) for every \(\varepsilon >0\) and \(N> N_0(c, \varepsilon)\). In 1952 \textit{I. I. Piatetski-Shapiro} [Mat. Sb., N. Ser.
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ON THE WARING–GOLDBACH PROBLEM FOR CUBES [PDF]
Glasgow Mathematical Journal, 2009 AbstractWe prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can
Brüdern, Jörg, Kawada, Koichi
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On the Hybrid Power Mean of Two‐Term Exponential Sums and Cubic Gauss Sums
Journal of Mathematics, Volume 2021, Issue 1, 2021., 2021 In this paper, an interesting third‐order linear recurrence formula is presented by using elementary and analytic methods. This formula is concerned with the calculating problem of the hybrid power mean of a certain two‐term exponential sums and the cubic Gauss sums.
Shaofan Cao +2 more
wiley +1 more source
Another Waring–Goldbach problem
Acta Arithmetica, 2023 In the paper under review, the author establishes a Waring-Goldbach analogue of a past result of C. Hooley. More specifically, \textit{C. Hooley} [Symp. Durham 1979, Vol. 1, 127--191 (1981; Zbl 0463.10037)] proved the theorem that for large \(n\), the Diophantine equation \[ x_1^2+x_2^2+x_3^3+x_4^4+x_5^5+x_6^6+x_7^7=n \tag{1} \] has solutions in non ...
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