Primes in Arithmetic Progression: The General Modulus

1980 ◽  
pp. 27-34
Author(s):  
Harold Davenport
Keyword(s):  
Arithmetic Progression ◽  
General Modulus

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).

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

Partitions into kth powers of terms in an arithmetic progression

2018 ◽  
Vol 290 (3-4) ◽  
pp. 1277-1307 ◽  
Author(s):  
Bruce C. Berndt ◽  
Amita Malik ◽  
Alexandru Zaharescu
Keyword(s):  
Arithmetic Progression

scholarly journals Rational tetrahedra with edges in arithmetic progression

2005 ◽  
Vol 111 (1) ◽  
pp. 57-80 ◽  
Author(s):  
C. Chisholm ◽  
J.A. MacDougall
Keyword(s):  
Arithmetic Progression

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

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