Volume 5, 1999, Number 2

Volume 5Number 1 ▷ Number 2 ▷ Number 3Number 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
Full paper (PDF, 385 Kb) | Abstract

R. D. Carmichael [1] conjectured that: “The equation 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 FA(k+1) , where FA(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

It has been suspected that if Fermat did indeed have a simple proof for his famous ‘last theorem’, that he probably employed his method of infinite descent. In a renewed attempt to see how Fermat might have thought that he had proved that if cn = an + bn, 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)

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