相关论文: Existence of $K$-multimagic squares and magic squa…

In this paper we investigate $K$-multimagic squares of order $N$, these are $N \times N$ magic squares which remain magic after raising each element to the $k$ th power for all $2 \leqslant$ $k \leqslant K$. Given $K \geqslant 2$, we…

A k-magic square of order n is an arrangement of the numbers from 0 to kn-1 in an n by n matrix, such that each row and each column has exactly k filled cells, each number occurs exactly once, and the sum of the entries of any row or any…

In this paper we give the first method for constructing n-multimagic squares (and hypercubes) for any n. We give an explicit formula in the case of squares and an effective existence proof in the higher dimensional case. Finally we prove…

For any $d \geq 2$, we prove that there exists an integer $n_0(d)$ such that there exists an $n \times n$ magic square of $d^\text{th}$ powers for all $n \geq n_0(d)$. In particular, we establish the existence of an $n \times n$ magic…

In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete…

It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same…

This paper aims to address the relation between a magic square of odd order $n$ and a group, and their properties. By the modulo number $n$, we construct entries for each table from initial table of magic square with large number $n^2$.…

A magic square of order n is an nxn square (matrix) whose entries are distinct nonnegative integers such that the sum of the numbers of any row and column is the same number, the magic constant. In this paper we introduce the concept of…

A magic square of order $n$ with all subsquares of possible orders (ASMS$(n)$) is a magic square which contains a general magic square of each order $k\in\{3, 4, \cdots, n-2\}$. Since the conjecture on the existence of an ASMS was proposed…

We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the…

Let $(\Gamma,+)$ be an Abelian group of order $n^2$ and MS$_{\Gamma}(n)$ be an $n\times n$ array whose entries are all elements of $\Gamma$. Then MS$_{\Gamma}(n)$ is a $\Gamma$-magic square if all row, column, main and backward main…

This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number of distinct squares in a finite word. By counting the number of (right-)special factors, we give an upper bound of the number of…

Magic squares are a fascinating mathematical challenge that has intrigued mathematicians for centuries. Given a positive (and possibly large) integer \( n \), one of the main challenges that still remains is to find, within a computational…

A magic series is a set of natural numbers that, by virtue of its size, sum, and maximum value, could fill a row of a normal magic square. In this paper, we derive an exact two-dimensional integral representation for the number of magic…

We explore some of the properties of consecutive, equally-summed arithmetic progressions of odd numbers, particularly their offsets and sums, before using them to prove that no $3\times3$ magic squares of distinct square integers exist.

A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an…

For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a…

Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of…