Volume 3, 1997, Number 1

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

On certain arithmetic functions associated with the unitary divisors of a number
Original research paper. Pages 1–8
József Sándor and László Tóth
Full paper (PDF, 254 Kb)

Negative subscript Jacobsthal numbers
Original research paper. Pages 9–22
A. F. Horadam
Full paper (PDF, 434 Kb)

A new continued fraction for Euler’s constant
Original research paper. Pages 23–34
Aldo Peretti
Full paper (PDF, 344 Kb)| Abstract

A new continued fraction is obtained for Euler constant C, namely:

    \[C = \frac{1}{2} - \frac{1}{3} = \Bigg \{ \frac{ \Big ( \frac{2r - 1}{r-1} \Big )^2 \mid }{ \mid \frac{-2^r + r}{r(r+1)} } + \frac{2^r + r \mid}{ \mid r^2} + r^2 \Big \{ \frac{(2^r + m)^2 \mid }{ \mid 1} \Big \}_{m=1}^{m=2^r - 1} \Bigg \}_{r = 2}^{r = \infty} \]

It is considered the possibility to prove the irrationality and transcendency of the constant by means of this expansion.

A curious problem involving geometric series
Original research paper. Pages 35–40
Piero Filipponi
Full paper (PDF, 190 Kb)

The anatomy of odd-exponent Diophantine triples
Original research paper. Pages 41–51
J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Full paper (PDF, 291 Kb) | Abstract

In similar manner to the analysis for even-exponent triples, this paper uses the equivalence classes of the modular ring ℤ6 to show why the diophantine equation dm = em + fm, where m is odd, is limited to m = 1 in ℤ6.

On some analogues of the Bourque–Ligh conjecture on LCM matrices
Original research paper. Pages 52–57
Pentti Haukkanen and Juha Sillanpää
Full paper (PDF, 249 Kb) | Abstract

Let S = {x_1, x_2, ..., x_n} be a set of distinct positive integers. The n \times n  matrix (S) whose i, j-entry is the greatest common divisor (x_i,x_j) of x_i and x_j is called the GCD matrix on S. The LCM matrix [S] on S is defined analogously. It is a direct consequence of a known determinant evaluation that the GCD matrix is always nonsingular on gcd-closed sets. Bourque and Ligh conjectured that the LCM matrix is always nonsingular on gcd-closed sets. It has been shown that this conjecture does not hold. In this paper we study certain analogues of this conjecture relating to GCD and LCM matrices on lcm-closed sets and some unitary analogues of GCD and LCM matrices.

A modification of A. Mullin’s inequality
Original research paper. Pages 58–60
Krassimir Atanassov and Stephan Danchev
Full paper (PDF, 107 Kb) | Abstract

A modification of one A. Mullin’s hypothesis is introduced and solved.

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

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