Results 41 to 50 of about 126,211 (116)
On pairs of equations with unequal powers of primes and powers of 2
AIMS MathematicsIt is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations, each containing two squares of primes, two cubes of primes, two biquadrates of primes, and $ 30 $ powers of 2.
Li Zhu
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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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An Invitation to Additive Prime Number Theory [PDF]
, 2005 2000 Mathematics Subject Classification: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number ...
Kumchev, A., Tolev, D.
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Sums of four squares of primes
, 2015 Let $E(N)$ denote the number of positive integers $n \le N$, with $n \equiv 4 \pmod{24}$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first
Kumchev, Angel V., Zhao, Lilu
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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.
M. Garaev
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On Waring–Goldbach problem involving fourth powers
Journal of Number Theory, 2011 It is proved that every sufficiently large positive integer \(N\equiv 13\pmod{240}\) can be represented as \(p_1^4+p_2^4+\dots +p_{12}^4+P^4\), where \(p_1,\dots,p_{12}\) are primes and \(P\) is a \(P_5\)-almost prime. For comparison we note that \textit{K. Kawada} and \textit{T. D. Wooley} [Proc. Lond. Math. Soc., III. Ser. 83, No. 1, 1--50 (2001; Zbl
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On the Waring–Goldbach problem for seventh powers [PDF]
Proceedings of the American Mathematical Society, 2005 We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of 46 46 seventh powers of prime numbers.
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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
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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 =
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What is the smallest prime? [PDF]
, 2012 What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.
Caldwell, Chris K., Xiong, Yeng
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