On some universal sums of generalized polygonal numbers

Fan Ge; Zhi-Wei Sun

Displaying similar documents to “On some universal sums of generalized polygonal numbers”

Universal forms $\sum a_{i} {(x_{i})}^{n}$ and Waring’s problem

L. Dickson (1936)

Acta Arithmetica

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A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${(α_{j})}_{j = 1}^{\infty}$ of scalars, there exists a subsequence ${(α_{k_{j}})}_{j = 1}^{\infty}$ such that either every subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series, or no subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series. In particular examples we decide which of the two cases holds.

Universal embeddings of $l^{1} (α)$ into the space of continuous functions on a product space

N. Kalamidas, Th. Zachariades (1989)

Colloquium Mathematicae

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The number of minimum points of a positive quadratic form

G. L. Watson

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CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

Multiplicatively dependent triples of Tribonacci numbers

Carlos Alexis Ruiz Gómez, Florian Luca (2015)

Acta Arithmetica

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We consider the Tribonacci sequence $T : = {T_{n}}_{n \geq 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n + 3} = T_{n + 2} + T_{n + 1} + T_{n}$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change $L$ -functions for $Gl (2)$ are non-negative at their center of symmetry.

Optimality of the Width- $w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Clemens Heuberger, Daniel Krenn (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider digit expansions $\sum_{j = 0}^{ℓ - 1} Φ^{j} (d_{j})$ with an endomorphism $Φ$ of an Abelian group. In such a numeral system, the $w$ -NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$ -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

Representation functions for binary linear forms

Fang-Gang Xue (2024)

Czechoslovak Mathematical Journal

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Let $ℤ$ be the set of integers, $ℕ_{0}$ the set of nonnegative integers and $F (x_{1}, x_{2}) = u_{1} x_{1} + u_{2} x_{2}$ be a binary linear form whose coefficients $u_{1}$ , $u_{2}$ are nonzero, relatively prime integers such that $u_{1} u_{2} \neq \pm 1$ and $u_{1} u_{2} \neq - 2$ . Let $f : ℤ \to ℕ_{0} \cup {\infty}$ be any function such that the set $f^{- 1} (0)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A, F} (n) = f (n)$ for all integers $n$ , where $r_{A, F} (n) = | {(a, a^{'}) : n = u_{1} a + u_{2} a^{'} : a, a^{'} \in A} |$ . We add the structure of difference for the binary linear form $F (x_{1}, x_{2})$ .

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

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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.

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ as a runs through $F^{\times}$ , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $| F^{\times} |$ ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

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Let $α > 1$ be irrational. Several authors studied the numbers $ℓ^{m} (α) = inf {| y | : y \in Λ_{m}, y \neq 0},$ where $m$ is a positive integer and $Λ_{m}$ denotes the set of all real numbers of the form $y = ϵ_{0} α^{n} + ϵ_{1} α^{n - 1} + \dots + ϵ_{n - 1} α + ϵ_{n}$ with restricted integer coefficients $| ϵ_{i} | \leq m$ . The value of $ℓ^{1} (α)$ was determined for many particular Pisot numbers and $ℓ^{m} (α)$ for the golden number. In this paper the value of $ℓ^{m} (α)$ is determined for irrational numbers $α$ , satisfying $α^{2} = a α \pm 1$ with a positive integer $a$ .