A Note Concerning Fermat Numbers

The Fermat’s Numbers are clearly all of the form 6n + 5, with one exception to this rule. This paper presents a general formula, showing the structure of Fermat numbers.

Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software

In accordance with the traits of parallel computing, the paper proposes a parallel algorithm to factorize the Fermat numbers through parallelization of a sequential algorithm. The kernel work to parallelize a sequential algorithm is presented by subdividing the computing interval into subintervals that are assigned to the parallel processes to perform the parallel computing. Maple experiments show that the parallelization increases the computational efficiency of factoring the Fermat numbers, especially to the Fermat number with big divisors.

The rise in maximum determinant matrix complexity

Introduction: In spite of the apparent relation between maximum determinant matrices of even (Hadamard matrices) and odd orders, the latter have particularly complex patterns of repetitive elements. This is what makes them unique and attractive for various applications in visual data processing, coding and masking. Purpose: Developing the theory of maximum determinant matrices, with the focus on using computer-aided analysis, and calculating unique matrices with unique pattern structures in their portraits. Results: We have found some peculiarities of maximum determinant matrices, outlined their families related to Fermat numbers, demonstrated the complication of patterns in other matrices as their orders grow. The presumption about the increasing complexity of structures as the matrix orders grow is confirmed by a chain of matrix portraits we demonstrate. As applied to orthogonal Belevitch matrices, it follows that they cannot be found even in small orders such as 66 or 86.

On the Diophantine Equation Fn = x^a \pm x^b \pm 1 in Mersenne and Fermat Numbers

In this article we investigate on the representation of Fibonacci numbers in the form x^a \pm x^b pm 1, for x in the sequence of Mersenne and Fermat numbers.

Circles on lattices and maximum determinant matrices

Introduction: The Hadamard conjecture about the existence of maximum determinant matrices in all orders multiple of 4 is closely related to Gauss's problem about the number of points with integer coordinates (Z3 lattice points) on a spheroid, cone, paraboloid or parabola. The location of these points dictates the number and types of extreme matrices. Purpose: Finding out how Gaussian points on sections of solids of revolution are related to the number and types of maximum determinant matrices with a fixed structure for odd orders. Specifying a precise upper bound of maximum determinant values for edged two-circulant matrices and the orders on which they prevail over simpler cyclic structures. Results: A newly proposed formula refines the overly optimistic Elich – Wojtas’ upper bound for the case of matrices with а fixed structure. Fermat numbers have a special role for orders of 4t + 1, and Barba numbers affect the formation of classes of maximum determinant matrices which occupy the areas of orders 4t + 3, successively replacing each other. For a two-circulant structure with an edge, the maximum order of an optimal symmetric solution is estimated as 67. It is proved that the determinant of edged block matrices is superior to the determinants of circulant matrices everywhere except for a special order 39. Practical relevance: Maximum (for a fixed structure) determinant matrices related to lattice points have a direct practical significance for noise-resistant coding, compression and masking of video data.