Results 11 to 20 of about 196 (47)
Sums of two squares and a power
, 2016 We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for $k\not\equiv 0 \bmod 4$ a
C. Hooley +10 more
core +1 more source
On Waring's problem: some consequences of Golubeva's method [PDF]
, 2012 We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the Hardy-Littlewood ...
Wooley, Trevor D.
core +2 more sources
On the Waring–Goldbach problem for seventh and higher powers [PDF]
, 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
core +3 more sources
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.
core +3 more sources
On the Waring problem for polynomial rings
, 2011 In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \in C[x_0,x_1,...,x_n] of degree divisible by k\ge 2 can be represented as a sum of at most k^n k-th powers
Fröberg, Ralf +2 more
core +2 more sources
, 2015 Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity.
Wooley, Trevor D.
core +1 more source
Sums of four prime cubes in short intervals
, 2018 We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known ...
Languasco, Alessandro +1 more
core +1 more source
Sum of one prime and two squares of primes in short intervals [PDF]
, 2015 Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.Comment: removed unconditional ...
Languasco, Alessandro +1 more
core +3 more sources
Almost all primes have a multiple of small Hamming weight
, 2016 Recent results of Bourgain and Shparlinski imply that for almost all primes $p$ there is a multiple $mp$ that can be written in binary as $mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k,$ with $k=66$ or $k=16$, respectively. We show that $
Elsholtz, Christian
core +1 more source
Relations between exceptional sets for additive problems [PDF]
, 2010 We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show,
Kawada, Koichi, Wooley, Trevor D.
core +3 more sources