**Volume 10** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Some properties of Fermatian numbers**

*Original research paper. Pages 25—33*

A. G. Shannon

Full paper (152 Kb) | Abstract

*n*-th reduced Fermatian number in terms of

so that 1* _{n}* =

*n*, and 1

*! =*

_{n}*n*!, where

*q*! =

_{n}*q*

_{n}q_{n−1}…

*q*

_{1}.

Some congruence properties and relationships with Bernoulli and Fibonacci numbers are explored. Some aspects of the notation and meaning of the Fermatian numbers are also outlined.

**On five Smarandache’s problems**

*Original research paper. Pages 34—53*

Mladen V. Vassilev-Missana and Krassimir T. Atanassov

Full paper (5956 Kb)

**Some representations concerning the product of divisors of n**

*Original research paper. Pages 54—56*

Mladen Vassilev-Missana

Full paper (896 Kb) | Abstract

*τ(n*) the number of all divisors of

*n*. It is well-known (see, e.g., ) that

(1)

and of course, we have

(2)

But (1) is not a good formula for *P _{d}*(

*n*), because it depends on function

*τ*and to express

*τ*(

*n*) we need the prime number factorization of

*n*.

Below, we give other representations of

*P*(

_{d}*n*) and

*p*(

_{d}*n*), which do not use the prime number factorization of

*n*.

**Volume 10** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4