Results 11 to 20 of about 268 (51)
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.
core +6 more sources
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
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
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
doaj +1 more source
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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PRIME SOLUTIONS TO POLYNOMIAL EQUATIONS IN MANY VARIABLES AND DIFFERING DEGREES
Forum of Mathematics, Sigma, 2018 Let $\mathbf{f}=(f_{1},\ldots ,f_{R})$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j ...
SHUNTARO YAMAGISHI
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Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
Open Mathematics, 2017 In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., N=p13+…+pj3$\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with |pi−(N ...
Feng Zhao
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A Diophantine approximation problem with two primes and one k-power of a prime [PDF]
, 2018 We refine a result of the last two Authors on a Diophantine approximation problem with two primes and a k-th power of a prime which was only proved to hold for ...
Alessandro, Gambini +2 more
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On Diophantine approximation by unlike powers of primes
Open Mathematics, 2019 Suppose that λ1, λ2, λ3, λ4, λ5 are nonzero real numbers, not all of the same sign, λ1/λ2 is irrational, λ2/λ4 and λ3/λ5 are rational. Let η real, and ε > 0.
Ge Wenxu, Li Weiping, Wang Tianze
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