Cyclotomic quadratic forms

François Sigrist

Displaying similar documents to “Cyclotomic quadratic forms”

Factors of a perfect square

Tsz Ho Chan (2014)

Acta Arithmetica

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We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between $\sqrt n - ∜ n {(l o g n)}^{1 / 7}$ and $\sqrt n + ∜ n {(l o g n)}^{1 / 7}$ .

On the computation of Hermite-Humbert constants for real quadratic number fields

Michael E. Pohst, Marcus Wagner (2005)

Journal de Théorie des Nombres de Bordeaux

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We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $ℚ (\sqrt{13})$ and $ℚ (\sqrt{17})$ . Finally we compute the Hermite-Humbert constant for the number field $ℚ (\sqrt{13})$ .

On the number of perfect binary quadratic forms.

Aicardi, Francesca (2004)

Experimental Mathematics

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Quasiperfect domination in triangular lattices

Italo J. Dejter (2009)

Discussiones Mathematicae Graph Theory

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A vertex subset S of a graph G is a perfect (resp. quasiperfect) dominating set in G if each vertex v of G∖S is adjacent to only one vertex ( $d_{v}$ ∈ 1,2 vertices) of S. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol 3,6 and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets S with induced components of the form $K_{ν}$ , where ν ∈ 1,2,3 depends only...

Perfect set properties in models of ZF

Carlos Augusto Di Prisco, Franklin C. Galindo (2010)

Fundamenta Mathematicae

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We study several perfect set properties of the Baire space which follow from the Ramsey property $ω \to {(ω)}^{ω}$ . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

Perfect set theorems

Otmar Spinas (2008)

Fundamenta Mathematicae

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We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for $G_{δ}$ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.

Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek (1980)

Annales Polonici Mathematici

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Abstract. Let F be a formally real field. Denote by G(F) and $G_{t} (F)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G (F) / G_{t} (F)$ . This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G (F) / G_{t} (F)$ , which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the...

On the index of an odd perfect number

Feng-Juan Chen, Yong-Gao Chen (2014)

Colloquium Mathematicae

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Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} | | N$ . Define the index $m = σ (N / q^{α}) / q^{α}$ . We prove that m cannot take the form $p^{2 u}$ , where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅,...

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets $A \subseteq_{p}$ which represent the quadratic residues $R \subseteq_{p}$ in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

Perfect unary forms over real quadratic fields

Dan Yasaki (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $F = ℚ (\sqrt{d})$ be a real quadratic field with ring of integers $𝒪$ . In this paper we analyze the number $h_{d}$ of ${GL}_{1} (𝒪)$ -orbits of homothety classes of perfect unary forms over $F$ as a function of $d$ . We compute $h_{d}$ exactly for square-free $d \leq 200000$ . By relating perfect forms to continued fractions, we give bounds on $h_{d}$ and address some questions raised by Watanabe, Yano, and Hayashi.

Binary quadratic forms and Eichler orders

Montserrat Alsina (2005)

Journal de Théorie des Nombres de Bordeaux

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For any Eichler order $𝒪 (D, N)$ of level $N$ in an indefinite quaternion algebra of discriminant $D$ there is a Fuchsian group $Γ (D, N) \subseteq SL (2, ℝ)$ and a Shimura curve $X (D, N)$ . We associate to $𝒪 (D, N)$ a set $ℋ (𝒪 (D, N))$ of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to $Γ (D, N)$ , for primitive forms contained in $ℋ (𝒪 (D, N))$ . In particular, the classification theory of primitive integral binary quadratic forms by $SL (2, ℤ)$ is recovered. Explicit fundamental domains for $Γ (D, N)$ allow the characterization...