Exceptional Set For Sums Research Articles

A note on the exceptional set for sums of unlike powers of primes

A note on the exceptional set for sums of unlike powers of primes

Exceptional Sets for Sums of Prime Cubes in Short Interval

Let E N , X denote the number of even integers n , with N − X ≤ n ≤ N , such that n cannot be written as n = p 1 3 + ⋯ + p 8 3 . We prove that if X > N 1 / 36 + ɛ , then E N , X = o X .

Slim exceptional set for sums of mixed powers of primes

Slim exceptional set for sums of mixed powers of primes

Exceptional set for sums of unlike powers of primes (II)

Let N be a sufficiently large integer. In this paper, it is proved that, with at most $$O(N^{7/18+\varepsilon })$$ exceptions, all even positive integers up to N can be represented in the form $$p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4$$ , where $$p_1,p_2,p_3,p_4,p_5,p_6$$ are prime numbers, which constitutes an improvement over some previous work.

Exceptional Set for Sums of Symmetric Mixed Powers of Primes

The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.

On exceptional sets of biquadratic Waring-Goldbach problem

TextIn this paper, we investigate exceptional sets in the biquadratic Waring-Goldbach problem. For example, we prove that all but O(N60/61) integers subject to the necessary local conditions can be represented as the sum of seven biquadrates of primes. Furthermore, the method in this paper can be easily used for exceptional sets for sums of fifth and higher of powers of primes. VideoFor a video summary of this paper, please visit https://youtu.be/PGa8P-Px8bk.

Slim exceptional set for sums of two squares, two cubes, and two biquadrates of primes

We prove that, with at most $$O(N^{\frac{17}{192}+\varepsilon})$$ exceptions, all even positive integers up to N are expressible in the form p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , where p1, p2, …, p6 are prime numbers. This gives large improvement of a recent result $$O(N^{\frac{13}{16}+\varepsilon})$$ due to M. Zhang and J. J. Li.

Exceptional sets for sums of almost equal prime cubes

Abstract In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

Exceptional set for sums of unlike powers of primes

Let [Formula: see text] denote the number of representations of an even integer [Formula: see text], [Formula: see text] as the sum of two squares, one cube and three fourth powers of primes, and by [Formula: see text] we denote the number of even integers [Formula: see text], [Formula: see text] such that the expected asymptotic formula for [Formula: see text] fails to hold. In this paper, it is proved that [Formula: see text] for any [Formula: see text].

Exceptional sets for sums of five and six almost equal prime cubes

We investigate the exceptional sets of natural number n which can be represented as sums of five and six cubes of almost equal primes, i.e. $${n=p_1^3+\cdots+p_s^3}$$ (s=5,6). It is established that almost all natural numbers n subject to certain congruence conditions have the above representation with $${|p_j-(n/s)^{\frac{1}{3}}|\leq n^{\theta_s/3+\varepsilon}}$$ ( $${1\leq j\leq s}$$ ), where $${\theta_5=8/9+\varepsilon}$$ and $${\theta_6=5/6+\varepsilon}$$ .

Exceptional Set for Sums of Unlike Powers of Primes

Let $N$ be a sufficiently large integer. In this paper, it is proved that with at most $O(N^{13/16+\varepsilon})$ exceptions, all even positive integers up to $N$ can be represented in the form $p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^4 + p_6^4$, where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$, $p_6$ are prime numbers.

Exceptional set for sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions can be written as sum of the form N = p 13 +…+ p 1 s with | p i( N/s )1/3| ≤ N 1/3- θ s , where θ s = 7/261 - 2 e , 5/159 - e , 11/333 - e , 19/561 - e for s = 5, 6, 7, 8, respectively.

The exceptional set for sums of unlike powers of primes

It is established that all even positive integers up to N but at most O(N15/16+ɛ) exceptions can be expressed in the form p12 + p23 + p34 + p45, where p1, p2, p3 and p4 are prime numbers.

On the Waring–Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

AbstractWe investigate exceptional sets in theWaring–Goldbach problem. For example, in the cubic case, we show that all but O(N79/84+∊) integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes.

Slim Exceptional Sets For Sums of Four Squares

Slim Exceptional Sets For Sums of Four Squares

Slim Exceptional Sets for Sums of Cubes

AbstractWe investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding X, that fail to have a representation as the sum of 7 cubes of prime numbers, is O(X23/36+ε). For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is O(X11/36+ε).

Slim exceptional sets for sums of fourth and fifth powers

Slim exceptional sets for sums of fourth and fifth powers