scholarly journals Waring–Goldbach Problem of Even Powers in Short Intervals

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Liqun Hu ◽  
Tanhui Zhang
Keyword(s):  
Prime Numbers ◽  
Short Intervals ◽  
Goldbach Problem ◽  
Average Behaviour

In this paper, we study the average behaviour of the representations of n = p 1 2 + p 2 4 + p 3 4 + p 4 k over short intervals for k ≥ 4 , where p 1 , p 2 , p 3 , p 4 are prime numbers. This improves the previous results.

scholarly journals Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

2017 ◽  
Vol 15 (1) ◽  
pp. 1517-1529
Author(s):  
Zhao Feng
Keyword(s):  
Short Intervals ◽  
Exceptional Sets ◽  
Goldbach Problem ◽  
Almost All

Abstract 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., $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

Waring-Goldbach problem for fourth powers in short intervals

2013 ◽  
Vol 8 (6) ◽  
pp. 1407-1423 ◽  
Author(s):  
Hengcai Tang ◽  
Feng Zhao
Keyword(s):  
Short Intervals ◽  
Goldbach Problem

On prime numbers of special kind on short intervals

2006 ◽  
Vol 79 (5-6) ◽  
pp. 848-853
Author(s):  
N. N. Mot’kina
Keyword(s):  
Special Kind ◽  
Prime Numbers ◽  
Short Intervals

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

Sums of Two Squares in Short Intervals

2000 ◽  
Vol 52 (4) ◽  
pp. 673-694 ◽  
Author(s):  
Antal Balog ◽  
Trevor D. Wooley
Keyword(s):  
Prime Numbers ◽  
Positive Dimension ◽  
Short Intervals ◽  
Sums Of Two Squares ◽  
Sieving Process

AbstractLet denote the set of integers representable as a sum of two squares. Since can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that hasmany properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of than expected, and infinitely many intervals containing considerably fewer than expected.

Exceptional Set in Waring–Goldbach Problem Involving Squares, Cubes and Sixth Powers

2021 ◽  
Vol 58 (1) ◽  
pp. 84-103
Author(s):  
Jinjiang Li ◽  
Min Zhang ◽  
Haonan Zhao
Keyword(s):  
Prime Numbers ◽  
Large Integer ◽  
Exceptional Set ◽  
Goldbach Problem ◽  
Positive Integers

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+s) exceptions, all even positive integers up to N can be represented in the form where p1, p2, p3, p4, p5, p6 are prime numbers.

scholarly journals Primes in explicit short intervals on RH

2016 ◽  
Vol 12 (05) ◽  
pp. 1391-1407 ◽  
Author(s):  
Adrian W. Dudek ◽  
Loïc Grenié ◽  
Giuseppe Molteni
Keyword(s):  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Short Intervals ◽  
The Difference

In this paper, on the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

scholarly journals Prime numbers of the form p=m2+n2+1 in short intervals

2007 ◽  
Vol 128 (2) ◽  
pp. 193-200 ◽  
Author(s):  
Kaisa Matomäki
Keyword(s):  
Prime Numbers ◽  
Short Intervals
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