**Volume 5** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On certain generalizations of the Smarandache function**

*Original research paper. Pages 41–51*

J. Sándor

Full paper (PDF, 240 Kb)

**An inequality concerning prime numbers**

*Original research paper. Pages 52–54*

L. Panaitopol

Full paper (PDF, 80 Kb)

**Carmichael’s conjecture and a minimal unique solution**

*Original research paper. Pages 55–70*

W. Ramadan-Jradi

Full paper (PDF, 385 Kb) | Abstract

*ϕ*(

*x*) =

*A*where

*ϕ*is Euler’s totient function, and

*A*is an even positive integer does not have a unique solution. This paper is a continue of paper [5], and it states some theorems and lemmas which help giving an “if and only if” condition for a unique solution, and locate this solution when it exists. By following the same approach of [5] we will be able to determine some particular properties of the set

*F*

_{A(k+1)}, where

*F*

_{A(k)}= {

*x*:

*ϕ*(

*x*) =

*A*

_{(k)}} as defined in [5] is a minimal set of solutions of the equation

*ϕ*(

*x*) =

*A*

_{(k)}.

**Speculation of Fermat’s proof of his Last Theorem**

*Original research paper. Pages 71–79*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 328 Kb) | Abstract

*c*=

^{n}*a*+

^{n}*b*,

^{n}*a*,

*b*,

*c*,

*n*∈ ℤ,

*n*> 2, then

*a*,

*b*,

*c*cannot all be integers, we set

*c*=

*a*+

*b*+

*m*,

*m*∈ ℤ, and then raised it to the

*n-*th power. The roots of the resulting polynomial in

*m*appear to be – (

*a*+

*b*) only when

*n*≠ 1 ,2 , and the result might have seemed to Fermat to follow from this. The plausibility of the algebra developed here is considered in the context of the work of the sixteenth century mathematicians, particularly Cardano and Bombelli.

**On the 22-nd, the 23-th and the 24-th Smarandache’s problems**

*Original research paper. Pages 80–82*

K. Atanassov

Full paper (PDF, 106 Kb)

**On the 37-th and 38-th Smarandache’s problems**

*Original research paper. Pages 83–85*

K. Atanassov

Full paper (PDF, 89 Kb)

**On the 43-rd and 44-th Smarandache’s problems**

*Original research paper. Pages 86–88*

Vassia K. Atanassova and Krassimir T. Atanassov

Full paper (PDF, 97 Kb)