相关论文: A New Transformation of the Magic Square of Square…

We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed…

We find the numbers of $3 \times 3$ magic, semimagic, and magilatin squares, as functions either of the magic sum or of an upper bound on the entries in the square. Our results on magic and semimagic squares differ from previous ones in…

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…

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…

Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…

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.

The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well known. We show how this correspondence can be extended to the…

Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a…

We generalize classical triangular Schubert puzzles to puzzles with convex polygonal boundary. We give these puzzles a geometric Schubert calculus interpretation and derive novel combinatorial commutativity statements, using purely…

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by…

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of…

Gaussian mixtures are a powerful and widely used tool to model non-Gaussian estimation problems. They are able to describe measurement errors that follow arbitrary distributions and can represent ambiguity in assignment tasks like point set…

Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define…

Magic-square constraints define Diophantine systems whose solutions, in several natural families, exhibit rigid periodic structure. We study this structure in an oracle setting, where a marked set of integers is given by black-box access…

A construction of the magic square, and hence of exceptional Lie algebras, is carried out using trialities rather than division algebras. By way of preparation, a comprehensive discussion of trialities is given, incorporating a number of…

It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately,…

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 square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the…

Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…

The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…