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Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Ebénézer Ntienjem (2017)

Open Mathematics

The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), ( l , m ) 0 2 α l + β m = n σ ( l ) σ ( m ) , where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by...

Expressing a number as the sum of two coprime squares.

Warren Dicks, Joan Porti (1998)

Collectanea Mathematica

We use hyperbolic geometry to study the limiting behavior of the average number of ways of expressing a number as the sum of two coprime squares. An alternative viewpoint using analytic number theory is also given.

Five regular or nearly-regular ternary quadratic forms

William C. Jagy (1996)

Acta Arithmetica

1. Introduction. In a recent article [6], the positive definite ternary quadratic forms that can possibly represent all odd positive integers were found. There are only twenty-three such forms (up to equivalence). Of these, the first nineteen were proven to represent all odd numbers. The next four are listed as "candidates". The aim of the present paper is to show that one of the candidate forms h = x² + 3y² + 11z² + xy + 7yz does represent all odd (positive) integers, and that it is regular in...

Kvaterniony a důkaz Lagrangeovy věty o čtyřech čtvercích

Matěj Doležálek (2019)

Pokroky matematiky, fyziky a astronomie

Článek představuje užití kvaternionů k důkazu Lagrangeovy věty o čtyřech čtvercích a použití stejných myšlenek k důkazům univerzálnosti dalších kvadratických forem. Užito je vlastností normy a ideálů v jistých kvaternionových oborech.

Legendre polynomials and supercongruences

Zhi-Hong Sun (2013)

Acta Arithmetica

Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), P [ p / 6 ] ( t ) - ( 3 / p ) x = 0 p - 1 ( ( x ³ - 3 x + 2 t ) / p ) ( m o d p ) and ( x = 0 p - 1 ( ( x ³ + m x + n ) / p ) ) ² ( ( - 3 m ) / p ) k = 0 [ p / 6 ] 2 k k 3 k k 6 k 3 k ( ( 4 m ³ + 27 n ² ) / ( 12 ³ · 4 m ³ ) ) k ( m o d p ) , where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning k = 0 p - 1 2 k k 3 k k 6 k 3 k / m k ( m o d p ² ) , where m is an integer not divisible by p.

Levels of rings - a survey

Detlev W. Hoffmann (2016)

Banach Center Publications

Let R be a ring with 1 ≠ 0. The level s(R) of R is the least integer n such that -1 is a sum of n squares in R provided such an integer exists, otherwise one defines the level to be infinite. In this survey, we give an overview on the history and the major results concerning the level of rings and some related questions on sums of squares in rings with finite level. The main focus will be on levels of fields, of simple noncommutative rings, in particular division rings, and of arbitrary commutative...

Minimal 𝒮 -universality criteria may vary in size

Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)

Journal de Théorie des Nombres de Bordeaux

In this note, we give simple examples of sets 𝒮 of quadratic forms that have minimal 𝒮 -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

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