Volume 6, 2000, Number 2

The aim of this article is to propose some remarks on the Bernoulli numbers. Firstly, a simple proof for the the equation B_{2n+ 1} = 0 is presented. This proof also gives an equation for ζ(2k). Using a simple computation, the values of ζ(2k), k = 1, 12 are presented. Finally, an equation for the infinite product  is proposed based on the Bernoulli numbers.

This article presents some new equations concerning the Euler function. An equation for the sum is found by using the multiplicative property. This is applied to find the sums and .

There exist cases when experiments with the computational part of the research could lead to new theoretical proofs. New hypotheses are suggested in concern with the Hardy–Littlewood conjecture that there exist infinitely many prime k-tuples. Although all the details on the presented investigation are given in another scope papers (applied mathematical logic), the formulation of the presented hypotheses has an independent role. Strong relations are revealed with the investigations from Number Theory period before the Hardy and Littlewood down to the Greek system. The introduced formulas help to establish upper and lower bounds for different constellations of prime k-tuples.

In the papers of Merlier [5] and Saito [7] there are given some necessary and sufficient conditions on the linear orderability of the bands. Similar questions on semigroups are treated by Jordjev, Todorov [4] and Todorov [9]. In the last paper there are studied for the first time the one-sided orderable semigroups. We considerably enlarge the last studies, giving conditions under which a given semigroup should not be linearly orderable, (being perhaps left or right stable orderable) and conditions when a semigroup is not one sided orderable, or when it is no-one-sided orderable.

The main results are: Theorem 1: For any semigroup S generated by the non-idempotent elements and the generating relations (i) ; (ii)   and (3) for   there exist non dual two-sided stable orders and (non dual) one sided stable orders which are not two sided stable ones and Theorem 2: Let S’ be a semigroup generated by the idempotents ,  the non-idempotents and the relations (i),(ii),(iii). Then the semigroup S’ has non-dual two-sided stable orders and only one-sided stable orders which are not two-sided stable ones.