Sums-of-Products and Subproblem Independence

2009 ◽  
pp. 301-325
Author(s):  
Richard E. Stearns ◽  
Harry B. Hunt
Keyword(s):  
Sums Of Products

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Exploring N-way Tables with Sums-of-Products Models

1993 ◽  
Vol 37 (3) ◽  
pp. 327-371 ◽  
Author(s):  
Jan P.H. Van Santen
Keyword(s):  
Sums Of Products

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).

New Expressions for Sums of Products of the Catalan Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 330
Author(s):  
Conghui Xie ◽  
Yuan He
Keyword(s):  
Arbitrary Number ◽  
Catalan Numbers ◽  
Bell Polynomials ◽  
Sums Of Products

In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. The results presented here can be regarded as the development of some known formulas.

scholarly journals SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES

Integers ◽  
2014 ◽  
Author(s):  
Emrah Kılıc ◽  
Nese Ömur
Keyword(s):  
Weighted Sums ◽  
Lucas Sequences ◽  
Sums Of Products

scholarly journals Stratification for Multiplicative Character Sums

2018 ◽  
Vol 2020 (10) ◽  
pp. 2881-2917 ◽  
Author(s):  
Junyan Xu
Keyword(s):  
Parameter Space ◽  
Rational Functions ◽  
Character Sums ◽  
Character Sum ◽  
Maximum Weight ◽  
Joint Work ◽  
Multiplicative Character ◽  
Multiplicative Characters ◽  
Sums Of Products ◽  
Multiplicative Character Sums

Abstract We prove a stratification result for certain families of n-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of r multiplicative characters evaluated at rational functions, and the families (with nr parameters) are obtained by allowing each of the r rational functions to be replaced by an “offset”, that is, a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least j⌊(r − 1)/2(n − 1)⌋ for 1 ≤ j ≤ n − 1 and ⌈nr/2⌉ for j = n. This result is used to obtain multivariate Burgess bounds in joint work with Lillian Pierce.

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