scholarly journals On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

2012 ◽  
Vol 24 (2) ◽  
pp. 355-368
Author(s):  
Maurizio Laporta
Keyword(s):  
Arithmetic Progression ◽  
Goldbach Problem ◽  
Prime Variable

scholarly journals Five squares in arithmetic progression over quadratic fields

2013 ◽  
Vol 29 (4) ◽  
pp. 1211-1238 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Xavier Xarles
Keyword(s):  
Arithmetic Progression ◽  
Quadratic Fields

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

Enlarged major arcs in the Waring–Goldbach problem

2016 ◽  
Vol 12 (01) ◽  
pp. 205-217 ◽  
Author(s):  
Taiyu Li
Keyword(s):  
Short Note ◽  
Circle Method ◽  
Goldbach Problem

In this short note, we treat the enlarged major arcs of circle method in the Waring–Goldbach problem.

Heronian Triangles with Sides in Arithmetic Progression: An Inradius Perspective

2012 ◽  
Vol 85 (4) ◽  
pp. 290-294 ◽  
Author(s):  
Herb Bailey ◽  
William Gosnell
Keyword(s):  
Arithmetic Progression

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.

scholarly journals Waring-Goldbach Problem: Two Squares and Three Biquadrates

2019 ◽  
Vol 23 (5) ◽  
pp. 1061-1071
Author(s):  
Yingchun Cai ◽  
Li Zhu
Keyword(s):  
Goldbach Problem
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