4-core partitions and class numbers
5-dissections and sign patterns of Ramanujan's parameter and its companion
In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction and its reciprocal. We obtain the 5-dissections for functions and , which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.
6 nombres dont les sommes deux à deux sont des carrés
A Banach space determined by the Weil height
A basis for the ring of doubly integer-valued polynomials.
A basis for the space of modular forms
A basis of ℤₘ, II
Given a set A ⊂ ℕ let denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and for all n̅ ∈ ℤₘ.
A basis of the set of sequences satisfying a given m-th order linear recurrence.
A basis of Zₘ
Let , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and for all n̅ ∈ Zₘ.
A Best Covering Problem.
A Best Lower Bound for Good Lattice Points.
A bijective proof of .
A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin.
A binomial coefficient identity associated with Beukers' conjecture on Apéry numbers.
A binomial representation of the 3x + 1 problem
A Bogomolov property for curves modulo algebraic subgroups
Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
A Bound for Prime Solutions of Some Ternary Equations.
A bound for the average rank of a family of abelian varieties
In this note, we consider a one-parameter family of Abelian varieties , and find an upper bound for the average rank in terms of the generic rank. This bound is based on Michel's estimates for the average rank in a one-parameter family of Abelian varieties, and extends previous work of Silverman for elliptic surfaces.
A bound for the discrepancy of digital nets and its application to the analysis of certain pseudo-random number generators
A bound for the first k-1 consecutive k-th power non-residues (mod p)