Results 1 to 10 of about 188 (37)
On the Waring-Goldbach problem for two squares and four cubes
Open Mathematics, 2023 Let NN be a sufficiently large integer. In this article, it is proved that, with at most O(N112+ε)O\left({N}^{\tfrac{1}{12}+\varepsilon }) exceptions, all even positive integers up to NN can be represented in the form p12+p22+p33+p43+p53+p63{p}_{1}^{2 ...
Zhang Min, Bai Hongxin, Li Jinjiang
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The Density of Rational Points on Non‐Singular Hypersurfaces, I [PDF]
, 2005 For any n ⩾ 3, let F ∈ Z[X0, …, Xn] be a form of degree d ⩾ 5 that defines a non‐singular hypersurface X ⊂ Pn. The main result in this paper is a proof of the fact that the number N(F; B) of Q‐rational points on X which have height at most B satisfies N ...
Tim D Browning, D. R. Heath-Brown
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Short intervals asymptotic formulae for binary problems with primes and powers, I: density 3/2 [PDF]
, 2016 We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square.
A., Zaccagnini, Languasco, Alessandro
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A pair of equations in unlike powers of primes and powers of 2
Open Mathematics, 2020 In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.
Cai Yong, Hu Liqun
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Equal sums of like polynomials [PDF]
, 2005 Let $f$ be a polynomial of degree $d>6$, with integer coefficients. Then the paucity of non-trivial positive integer solutions to the equation $f(a)+f(b)=f(c)+f(d)$ is established.
Browning, T. D.
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Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
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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Exceptional sets in Waring's problem: two squares and s biquadrates [PDF]
, 2014 Let $R_s(n)$ denote the number of representations of the positive number $n$ as the sum of two squares and $s$ biquadrates. When $s=3$ or $4$, it is established that the anticipated asymptotic formula for $R_s(n)$ holds for all $n\le X$ with at most $O(X^
Zhao, Lilu
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On pairs of equations in unlike powers of primes and powers of 2
Open Mathematics, 2017 In this paper, we obtained that when k = 455, every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of unlike powers of primes and k powers of 2.
Hu Liqun, Yang Li
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On the Waring--Goldbach problem for eighth and higher powers [PDF]
, 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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On Waring's problem for intermediate powers [PDF]
, 2016 Let $G(k)$ denote the least number $s$ such that every sufficiently large natural number is the sum of at most $s$ positive integral $k$th powers. We show that $G(7)\le 31$, $G(8)\le 39$, $G(9)\le 47$, $G(10)\le 55$, $G(11)\le 63$, $G(12)\le 72$, $G(13 ...
Wooley, Trevor D.
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