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.

ON THE REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2012 ◽  
Vol 09 (01) ◽  
pp. 189-204 ◽  
Author(s):  
ERNEST X. W. XIA ◽  
OLIVIA X. M. YAO
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Quadratic Forms ◽  
Theta Functions ◽  
Explicit Formulas ◽  
Representations Of Integers ◽  
Theta Function Identities

In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit formulas are obtained for the number of representations of a positive integer n by the quadratic forms [Formula: see text] with a ≠ 0, a + b + c = 4 and [Formula: see text] with k + l = 2 and r + s + t = 2 by employing these identities.

Evaluation of the convolution sums ∑a1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) with lcm(a1,a2,a3,a4) ≤ 4

2017 ◽  
Vol 13 (08) ◽  
pp. 2155-2173
Author(s):  
Joohee Lee ◽  
Yoon Kyung Park
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Divisor Functions ◽  
Positive Integers

The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums [Formula: see text] for the positive integers [Formula: see text], and [Formula: see text] with lcm[Formula: see text]. We reprove the known formulas for the number of representations of a positive integer [Formula: see text] by each of the quadratic forms [Formula: see text] as an application of the new identities proved in this paper.

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.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

scholarly journals Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

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.

FOURTEEN OCTONARY QUADRATIC FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 37-50 ◽  
Author(s):  
AYŞE ALACA ◽  
ŞABAN ALACA ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Sum Of Divisors ◽  
Recent Evaluation

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

scholarly journals CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

scholarly journals Representations by octonary quadratic forms with coefficients 1, 2, 3 or 6

2017 ◽  
Vol 13 (03) ◽  
pp. 735-749 ◽  
Author(s):  
Ayşe Alaca ◽  
M. Nesibe Kesicioğlu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms

Using modular forms, we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

scholarly journals Eichler Maps and Hyperbolic Fourier Expansion

1970 ◽  
Vol 40 ◽  
pp. 173-192 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Fourier Expansion ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Ternary Quadratic Forms ◽  
Hilbert Modular ◽  
Lecture Notes

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

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