scholarly journals The Bases of and the Number of Representation of Integers

2013 ◽  
Vol 2013 ◽  
pp. 1-34
Author(s):  
Barış Kendirli
Keyword(s):  
Quadratic Forms ◽  
Fundamental Theorem ◽  
Class Group ◽  
Equivalence Classes ◽  
Explicit Formulas ◽  
Direct Sums ◽  
Representation Of Integers ◽  
Positive Integers

Following a fundamental theorem of Hecke, some bases of and are determined, and explicit formulas are obtained for the number of representations of positive integers by all possible direct sums (111 different combinations) of seven quadratic forms from the class group of equivalence classes of quadratic forms with discriminant −71 whose representatives are .

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.

Summation of Singular Series Corresponding to the Representation of Integers by Some Quadratic Forms in Fourteen Variables

2000 ◽  
Vol 7 (2) ◽  
pp. 355-372
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Singular Series ◽  
Representation Of Integers ◽  
Positive Integers

Abstract A sum of the singular series corresponding to the number of representations of positive integers by some diagonal quadratic forms with integral coefficients is obtained.

LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

2007 ◽  
Vol 03 (03) ◽  
pp. 475-501 ◽  
Author(s):  
THOMAS A. SCHMIDT ◽  
MARK SHEINGORN
Keyword(s):  
Quadratic Forms ◽  
Closed Geodesics ◽  
Modular Surface ◽  
Explicit Formulas ◽  
Indefinite Quadratic ◽  
Positive Integers ◽  
Simple Closed Geodesics ◽  
Hall Ray

The Markoff spectrum of binary indefinite quadratic forms can be studied in terms of heights of geodesics on low-index covers of the modular surface. The lowest geodesics on [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. In our previous work, we classified the low height-achieving non-simple geodesics of [Formula: see text] into seven types according to the topology of highest arcs. Here, we obtain explicit formulas for the heights of geodesics of the first three types; the conjecture holds for approximation by closed geodesics of any of these types. Explicit examples show that each of the remaining types is realized.

On the Number of Representations of Positive Integers by a Direct Sum of Binary Quadratic Forms with Discriminant –23

1997 ◽  
Vol 4 (6) ◽  
pp. 523-532
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Binary Quadratic Forms ◽  
Direct Sums ◽  
Direct Sum ◽  
Positive Integers

Abstract Explicit exact formulas are obtained for the number of representations of positive integers by some direct sums of quadratic forms and .

On the Representation of Numbers by the Direct Sums of Some Quaternary Quadratic Forms

2001 ◽  
Vol 8 (1) ◽  
pp. 87-95
Author(s):  
N. Kachakhidze
Keyword(s):  
Quadratic Forms ◽  
Cusp Forms ◽  
Arbitrary Integer ◽  
Direct Sums ◽  
Positive Integers

Abstract The systems of bases are constructed for the spaces of cusp forms S 2m (Γ0(5), χ m ) and S 2m (Γ0(13), χ m ) for an arbitrary integer m ≥ 2. Formulas are obtained for the number of representations of a positive integers by the direct sums of some quaternary quadratic forms.

On Some Entire Modular Forms of Weights and for the Congruence Group Γ0(4N)

1996 ◽  
Vol 3 (5) ◽  
pp. 485-500
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Group ◽  
Positive Integers

Abstract Entire modular forms of weights and for the congruence group Γ0(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 7 and 9 variables.

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

scholarly journals Representation of integers: a nonclassical point of view

2020 ◽  
Vol 12 ◽  
Author(s):  
Bellaouar Djamel ◽  
Boudaoud Abdelmadjid
Keyword(s):  
Point Of View ◽  
Survey Article ◽  
Representation Of Integers ◽  
Positive Integers

In \cite{A.Boudaoud1}, the author asked the following question: Which $n\in \mathbb{N}$ unlimited can be represented as a sum $% n=s+w_{1}w_{2}$, where $s\in \mathbb{Z}$ is a limited integer and $\omega _{1}$, $\omega _{2}$ are two unlimited positive integers? In this survey article we partially answer this question, i.e., we present some families of unlimited positive integers which can be written as the sum of a limited integer and the product of at least two unlimited positive integers.

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