On Square-Free Numbers

Grabowski, Adam

doi:10.2478/forma-2013-0017

Cited by 6 publications

(2 citation statements)

References 10 publications

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“…The digits sum or digital sum of integers has been a subject of interest because of its application in cryptography, primality testing, random number generation and data reduction. Details on the origin, theories and applications of the digits sum of squared positive integers, integers and other important number sequences can be found in [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] . Recently digits sum and digital root have been applied in the analysis of lotto results [29] .…”

Section: Experimental Design Materials and Methodsmentioning

confidence: 99%

Datasets on the statistical properties of the first 3000 squared positive integers

et al. 2017

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The data in this article are as a result of a quest to uncover alternative research routes of deepening researchers’ understanding of integers apart from the traditional number theory approach. Hence, the article contains the statistical properties of the digits sum of the first 3000 squared positive integers. The data describes the various statistical tools applied to reveal different statistical and random nature of the digits sum of the first 3000 squared positive integers. Digits sum here implies the sum of all the digits that make up the individual integer.

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Section: Experimental Design Materials and Methodsmentioning

confidence: 99%

Datasets on the statistical properties of the first 3000 squared positive integers

et al. 2017

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“…The c 2015 University of Białystok current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11]. The notation and terminology used in this paper have been introduced in the following articles: [1], [25], [2], [3], [9], [26], [23], [4], [13], [28], [20], [14], [8], [5], [27], and [7].…”

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confidence: 99%

Stone Lattices

Grabowski

2015

Formalized Mathematics

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Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

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The group of squarefree integers

Hilberdink¹

2014

Linear Algebra and its Applications

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On Square-Free Numbers

Cited by 6 publications

References 10 publications

Datasets on the statistical properties of the first 3000 squared positive integers

Datasets on the statistical properties of the first 3000 squared positive integers

Stone Lattices

The group of squarefree integers

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