Volume 16, 2010, Number 4

A complete demonstration of solutions of the above Diophantine equation is given when are primes and are positive integers. Among the several examples exhibited, Example 3 provides a new solution containing eighty-five even numbers all of which are of the required form. Certain questions and modifications of the equation are also discussed.

Structural constraints prevent the difference of two odd cubes ever equaling an even cube. This is illustrated from the row structure of the modular ring Z₄. The critical structure factor is that the rows of integers, N², with 3 | N², follow the triangular numbers, whereas 3 ∤ N² rows follow the pentagonal numbers. This structural characteristic is the reason for the importance of primitive Pythagorean triples (in which either the smallest odd component or the even component always has a factor 3).

Some results of Gould for the ordinary Bernoulli and Euler polynomials are extended to analogous results built upon the Fermatian exponentials.

Two new sequences from Fibonacci type are introduced and the explicit formulae for their n-th members are given.

An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. Similarly, a vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. We prove that both rc(G) and rvc(G) have sharp concentration in classical random graph model G(n, p).

In the present paper some new results, concerning multiplicative functions with strictly positive values, are obtained. In particular, it is shown that if an ordered pair of such functions (f, g) has a certain property (called in the paper S), then for every fixed positive integer n; the minimal and the maximal elements of the set {f(d)g(n/d) : d runs over all divisors of n} are obtained at least for some unitary divisors of n. For these divisors if the maximum of f(d)g(n/d) is reached for d*; then the minimum is reached for n/d* and vice versa (the main results here are Theorems 1-4). The same investigation is made, but when d runs over the set of all divisors of n different than 1 and n (the main result here is Theorem 5). Also corollaries of the mentioned results are obtained and some particular cases are considered.