Volume 16, 2010, Number 2

An improved solution of the title equation with k = 52, x₅₂ = 1963 is exhibited. This is the best known result thus far.

Let a⊕b = max(a, b) and a⊗b = a+b for a, b ∈ ℝ and extend the pair of operations to matrices and vectors in the same way as in linear algebra. The homogeneous twosided system in max-algebra is of the form A ⊗ x = B ⊗ x. No polynomial method for solving homogeneous system is known. In this paper, we consider homogeneous twosided linear systems in max-algebra in a special case. We show that it can be checked in O(n³) time whether a given two-sided homogeneous system belongs to this special case. Solvability can be decided in O(n³) time and in the positive case a solution can be found in O(n³).

Let f(X₁, …, X_m) be a quadratic form in m variables X₁, …, X_m with integer coefficients. Then it is well-known that the Diophantine equation f(X₁, …, X_m) = 0 has a nontrivial solution in integers if and only if the equation has a nontrivial solution in real numbers and the congruence f(X₁, …, X_m) ≡ 0 (mod N) has a nontrivial solution for every integer N > 1. Such a principle is called the Hasse principle. In this paper, we explicitly give several types of families of the Diophantine equations of degree two, not homogeneous, for which the Hasse principle fails.

Simple functions were obtained for the rows of odd powers in the modular ring Z₄, wherein integer N is represented by N = 4r_i + i, i = 0, 1, 2, 3. These row functions are based on the row functions for squares. When 3 ∤ N , the row of N² = 3n(3n ±1) , or when 3|N, the row of N² = 2 + 18(½n(n+1)), n = 0, 1, 2, 3…

This note fleshes out some of the characteristics of what is referred to as a Zeckendorf triangle which is composed of Fibonacci number multiples of the Fibonacci sequence. It arose it arose in an infinite binary matrix related to the Zeckendorf representations of the non-negative integers.