Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

EDUCAÇÃO INTEGRAL: IDEAL ACALENTADO E CONCEITO MALTRATADO

O ensaio objetiva discutir formas de apropriação conceitual do ideal de educação integral. A partir do método de revisão bibliográfica, contemplando referências históricas do pensamento pedagógico, a legislação educacional brasileira e a literatura educacional brasileira contemporânea, busca-se apreender em que consiste a educação integral. Os resultados evidenciam que não há homogeneidade na compreensão da educação integral, embora, paradoxalmente, exista consenso em sua busca. Além da necessidade de maior aprofundamento conceitual e contextualização histórica, as elaborações em torno da educação integral são muito divergentes e não aprofundam o seu sentido, fornecendo balizas superficiais para políticas públicas e assimilando-a a políticas de proteção social. Por fim, o debate sobre o tema reivindica a fundamentação a partir de uma filosofia do ser humano que discuta, efetivamente, a integralidade de suas dimensões constitutivas e permita sua expressão em termos de necessidades educativas.

On a problem of Hasse and Ramachandra

Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integer N excluding 2, 3, 4 and 6.

On the least prime ideal and Siegel zeros

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

Overrings of Prüfer v-multiplication domains

Let [Formula: see text] be an integral domain, [Formula: see text] and [Formula: see text] the set of fractional ideals of [Formula: see text]. Let [Formula: see text] a finitely generated ideal with [Formula: see text]. For a torsion-free [Formula: see text]-module [Formula: see text], define [Formula: see text] for some [Formula: see text]. Call [Formula: see text] a [Formula: see text]-module if [Formula: see text]. On [Formula: see text], the function [Formula: see text] is a star-operation of finite character. An integral ideal [Formula: see text] maximal with respect to being a proper [Formula: see text]-ideal is a prime ideal called a maximal [Formula: see text]-ideal. A torsion-free [Formula: see text]-module [Formula: see text] is called [Formula: see text]-flat, if [Formula: see text] is a flat [Formula: see text]-module for each [Formula: see text], the set of maximal [Formula: see text]-ideals of [Formula: see text]. [Formula: see text] is called a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD), if [Formula: see text] is a valuation ring for each [Formula: see text]. We characterize [Formula: see text]-flat modules in a manner similar to the characterization of flat modules, study them when they are rings [Formula: see text] with [Formula: see text] and characterize P[Formula: see text]MDs using them and compare our work with similar work in the literature.

A FAMILY OF ASYMPTOTICALLY GOOD LATTICES HAVING A LATTICE IN EACH DIMENSION

A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n ≥ 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field ℚ(ζq) where q is the smallest prime congruent to 1 modulo n.