Representations of Integers as Sums of Nonvanishing Squares

1985 ◽  
pp. 72-83
Author(s):  
Emil Grosswald
Keyword(s):  
Representations Of Integers

On the Number of Representations of Integers by Quadratic Forms in Twelve Variables

1998 ◽  
Vol 5 (6) ◽  
pp. 545-564
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Explicit Formulas ◽  
Representations Of Integers ◽  
Positive Integers

Abstract A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.

Multiplicative representations of integers

1987 ◽  
Vol 57 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Representations Of Integers

Additive representations of integers

1998 ◽  
Vol 95 (3) ◽  
pp. 311-321
Author(s):  
Takashi Agoh
Keyword(s):  
Representations Of Integers

69.34 Representations of Integers and Permutations

1985 ◽  
Vol 69 (450) ◽  
pp. 283
Author(s):  
N. D. Thomson
Keyword(s):  
Representations Of Integers

The number of representations of integers by 4-dimensional strongly N-modular lattices

2021 ◽  
Author(s):  
Ho Yun Jung ◽  
Chang Heon Kim ◽  
Kyoungmin Kim ◽  
Soonhak Kwon
Keyword(s):  
Modular Lattices ◽  
Representations Of Integers

scholarly journals On the number of primitive representations of integers as sums of squares

2007 ◽  
Vol 13 (1-3) ◽  
pp. 7-25 ◽  
Author(s):  
Shaun Cooper ◽  
Michael Hirschhorn
Keyword(s):  
Sums Of Squares ◽  
Representations Of Integers

scholarly journals ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2008 ◽  
Vol 78 (1) ◽  
pp. 129-140 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Triangular Numbers ◽  
Representations Of Integers

AbstractGenerating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

scholarly journals Interesting identities involving weighted representations of integers as sums of arbitrarily many squares

2019 ◽  
Vol 116 (39) ◽  
pp. 19374-19379
Author(s):  
Min-Joo Jang ◽  
Ben Kane ◽  
Winfried Kohnen ◽  
Siu Hang Man
Keyword(s):  
Modular Forms ◽  
Analytic Functions ◽  
Sum Of Squares ◽  
Weighted Sums ◽  
Real Analysis ◽  
Representations Of Integers

We consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.

The number of representations of integers by generalized Bell ternary quadratic forms

2020 ◽  
pp. 1-15
Author(s):  
Kyoungmin Kim ◽  
Yeong-Wook Kwon
Keyword(s):  
Quadratic Form ◽  
Class Number ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Cusp Forms ◽  
Ternary Quadratic Form ◽  
Closed Formula ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Representations Of Integers

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

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