Elementary Proofs of Various Facts about 3-cores
نویسندگان
چکیده
Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n + 1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n ≥ 0, a3(4n + 1)= a3(n). 2000 Mathematics subject classification: primary 05A17, 11P81, 11P83.
منابع مشابه
Landau's Theorem revisited
Two new elementary proofs are given of Landau's Theorem on necessary and sufficient conditions for a sequence of integers to be the score sequence for some tournament. The first is related to existing proofs by majorization, but it avoids depending on any facts about majorization. The second is natural and direct. Both proofs are constructive, so they each provide an algorithm for obtaining a t...
متن کاملAn Ω-logic Primer
In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ωlogic and the Ω-conjecture.
متن کاملChevalley Groups over Commutative Rings I. Elementary Calculations
This is the rst in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar result...
متن کاملA short elementary proof of the Lagrange multiplier theorem
We present a short elementary proof of the Lagrange multiplier theorem for equality-constrained optimization. Most proofs in the literature rely on advanced analysis concepts such as the implicit function theorem, whereas elementary proofs tend to be long and involved. By contrast, our proof uses only basic facts from linear algebra, the definition of differentiability, the critical-point condi...
متن کاملAutomating Elementary Number-Theoretic Proofs Using Gröbner Bases
We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain number-theoretic predicates such as ‘divisible by’, ‘congruent’ and ‘coprime’; one notable example in this class is the Chinese Remaind...
متن کامل