scholarly journals On Multiplicative Magic Squares

10.37236/457 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Javier Cilleruelo ◽  
Florian Luca
Keyword(s):  
Lower Bound ◽  
Minimal Element ◽  
Magic Squares ◽  
Magic Square ◽  
Positive Integers

In this note, we give a lower bound for the distance between the maximal and minimal element in a multiplicative magic square of dimension $r$ whose entries are distinct positive integers.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

Magic squares of squares over a finite field

2021 ◽  
pp. 111-122
Author(s):  
Stewart Hengeveld ◽  
Giancarlo Labruna ◽  
Aihua Li
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Similar Question ◽  
Magic Squares ◽  
Content Type ◽  
Magic Square ◽  
Twin Primes ◽  
Quadratic Residues ◽  
Perfect Squares ◽  
Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

scholarly journals Magic square and Dirac flavor neutrino mass matrix

2020 ◽  
Vol 35 (29) ◽  
pp. 2050183
Author(s):  
Yuta Hyodo ◽  
Teruyuki Kitabayashi
Keyword(s):  
Neutrino Mass ◽  
Mass Matrix ◽  
Neutrino Mass Matrix ◽  
Magic Squares ◽  
Magic Square ◽  
Normal Mass

The magic texture is one of the successful textures of the flavor neutrino mass matrix for the Majorana type neutrinos. The name “magic” is inspired by the nature of the magic square. We estimate the compatibility of the magic square with the Dirac, instead of the Majorana, flavor neutrino mass matrix. It turned out that some parts of the nature of the magic square are appeared approximately in the Dirac flavor neutrino mass matrix and the magic squares prefer the normal mass ordering rather than the inverted mass ordering for the Dirac neutrinos.

scholarly journals Some results on magic squares based on generating magic vectors and R-C similar transformations

2017 ◽  
Vol 5 (1) ◽  
pp. 82-96 ◽  
Author(s):  
Xiaoyang Ma ◽  
Kai-tai Fang ◽  
Yu hui Deng
Keyword(s):  
Transformation Method ◽  
New Method ◽  
Magic Squares ◽  
Magic Square ◽  
Similar Transformation

Abstract In this paper we propose a new method, based on R-C similar transformation method, to study classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging two rows and related two columns of a magic square. Many new results for classification of the magic squares of order 5 are obtained by the R-C similar transformation method. Relationships between basic forms and R-C similar magic squares are discussed. We also propose a so called GMV (generating magic vector) class set method for classification of magic squares of order 5, presenting 42 categories in total.

Magic squares: would you believe…?

1974 ◽  
Vol 21 (5) ◽  
pp. 439-441
Author(s):  
David L. Pagni
Keyword(s):  
Main Diagonal ◽  
Magic Squares ◽  
Magic Square ◽  
Square Array ◽  
Magic Constant

A magic square of nth order is a square array of n rows and n columns whose components are n^2 distinct integers. Furthermore, the sum of the numbers in any row, column, or main diagonal must always equal a constant—the “magic constant.– The array in figure I, then, is a magic square of 3rd order whose magic constant is the number 15.

Constructing Magic Square Number Games

1978 ◽  
Vol 26 (2) ◽  
pp. 36-38
Author(s):  
John E. Bernard
Keyword(s):  
Magic Squares ◽  
Educational Value ◽  
Magic Square

How often have you seen children fill page after page with tic-tac-toe games and hoped for a way to direct their energy and enthusiasm into the learning of mathematics? This article describes how magic squares can be used to generate an assortment of number games that are “the same as” tic-tac-toe. Perhaps these games will be prized not only for their educational value, but also because they provide tic-tac-toe with stiff competition in the “interest- getting department.”

The Odd-numbered Magic Square

1989 ◽  
Vol 82 (2) ◽  
pp. 139-141
Author(s):  
James X. Paterno
Keyword(s):  
Elementary School ◽  
Magic Squares ◽  
Magic Square ◽  
Square Configuration

Acommon exercise in elementary school arithmetic is the completion of magic squares, in which the pupil is to enter certain numerals so that the sum of any row, column, or diagonal is constant. The simplest form involves the use of numerals one through n2, where n2 represents the total number of boxes in a square configuration n boxes by n boxes. This article reviews a familar procedure for generating magic squares and points out a surprising pattern.

The Versatile Magic Square

2003 ◽  
Vol 8 (5) ◽  
pp. 252-255
Author(s):  
Gale A. Watson
Keyword(s):  
Special Needs ◽  
Elementary Grades ◽  
Magic Squares ◽  
Magic Square

Article demonstrates the transformations that are possible to construct a variety of magic squares, including modifications to challenge students from elementary grades through algebra. An example of using magic squares with students who have special needs is also shared.

scholarly journals Monochromatic sequences whose gaps belong to {d, 2d, …, md}

1998 ◽  
Vol 58 (1) ◽  
pp. 93-101 ◽  
Author(s):  
Bruce M. Landman
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Order Of Magnitude ◽  
Positive Integers

For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} such that for some positive integer d, xi + 1 − xi ∈ {d, 2d,…, md} for i = 1,…, k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2, …, n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r ≤ m, hm(m + r) ≤ 2c(m + r − 1) + 1, where c = ⌈m/(m − r)⌉. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

scholarly journals Vector Spaces of New Special Magic Squares: Reflective Magic Squares, Corner Magic Squares, and Skew-Regular Magic Squares

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Thitarie Rungratgasame ◽  
Pattharapham Amornpornthum ◽  
Phuwanat Boonmee ◽  
Busrun Cheko ◽  
Nattaphon Fuangfung
Keyword(s):  
Linear Algebra ◽  
Standard Addition ◽  
Scalar Multiplication ◽  
Vector Spaces ◽  
Magic Squares ◽  
Magic Square ◽  
Definition Of

The definition of a regular magic square motivates us to introduce the new special magic squares, which are reflective magic squares, corner magic squares, and skew-regular magic squares. Combining the concepts of magic squares and linear algebra, we consider a magic square as a matrix and find the dimensions of the vector spaces of these magic squares under the standard addition and scalar multiplication of matrices by using the rank-nullity theorem.

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