Results 1 to 10 of about 275 (62)
Vinogradov systems with a slice off [PDF]
Mathematika, Volume 63, Issue 3, Page 797-817, 2017., 2017 Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{$1\le j\le k$, $j\ne r$}),$$ with $1\le x_i,y_i\le X$ $(1\le i\le s)$.
Brandes, Julia, Wooley, Trevor D.
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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
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Diophantine approximation with one prime, two squares of primes and one kth power of a prime
Open Mathematics, 2021 Let ...
Gambini Alessandro
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Sums of four and more unit fractions and approximate parametrizations
Bulletin of the London Mathematical Society, Volume 53, Issue 3, Page 695-709, June 2021., 2021 Abstract We prove new upper bounds on the number of representations of rational numbers mn as a sum of four unit fractions, giving five different regions, depending on the size of m in terms of n. In particular, we improve the most relevant cases, when m is small, and when m is close to n.
Christian Elsholtz, Stefan Planitzer
wiley +1 more source
RATIONAL CURVES ON CUBIC HYPERSURFACES OVER FINITE FIELDS
Mathematika, Volume 67, Issue 2, Page 366-387, April 2021., 2021 Abstract Given a smooth cubic hypersurface X over a finite field of characteristic greater than 3 and two generic points on X, we use a function field analogue of the Hardy–Littlewood circle method to obtain an asymptotic formula for the number of degree d k‐rational curves on X passing through those two points.
Adelina Mânzăţeanu
wiley +1 more source
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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AN AVERAGE THEOREM FOR TUPLES OF k‐FREE NUMBERS IN ARITHMETIC PROGRESSIONS
Mathematika, Volume 67, Issue 1, Page 1-35, January 2021., 2021 Abstract In the spirit of the Hooley–Montgomery refinement of the Barban–Davenport‐Halberstam theorem, we obtain an asymptotic formula for the variance associated with tuples of k‐free numbers in arithmetic progressions.
Tomos Parry
wiley +1 more source
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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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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