Volume 5, 1999, Number 3

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


On the 97-th, the 98-th and 99-th Smarandache’s problems
Original research paper. Pages 89—93
H. Aladjov and K. Atanassov
Full paper (PDF, 145 Kb)

On the 100-th, the 101-st and 102-nd Smarandache’s problems
Original research paper. Pages 94—96
K. Atanassov
Full paper (PDF, 93 Kb)

On the 117-th Smarandache’s problem
Original research paper. Pages 97—98
K. Atanassov
Full paper (PDF, 51 Kb)

On the 118-th Smarandache’s problem
Original research paper. Page 99
K. Atanassov
Full paper (PDF, 39 Kb)

On the 62-nd Smarandache’s problem
Original research paper. Pages 100—101
K. Atanassov
Full paper (PDF, 78 Kb)

Analyses of row expansions within the octic “chess” modular ring, Z8
Original research paper. Pages 102—114
J. Leyendekkers and A. Shannon
Full paper (PDF, 558 Kb) | Abstract

#! A modular ring Z8is described as an expanded form of Z4 in which the parities of the rows within the four classes of Z4 are distinguished. This octic or ‘chess’ ring is used to analyse thestructure of rows characterising powers x” within the eight classes of Z8. With n even, three classes
contain the squares, x2, but only two contain x” when n > 2. With n odd, even x” are confined to 0g, but odd numbers, x”, have the same class as x. The rows containing squares are related to first order recurrences, while rows containing x”, n > 2, are related to third and higher order recurrences. Because of this it is conjectured that the sum or difference of x”, yn cannot fit into a row which is characteristic of a power and hence cannot equal a power when n > 2.

A simple proof that the area of a Pythagorean triangle is square-free
Original research paper. Pages 115—118
J. Leyendekkers and A. Shannon
Full paper (PDF, 157 Kb) | Abstract

By using the geometry of a pythagorean triangle with a circle inscribed, it can be proved by a simple geometric proof that the area of such a triangle can never be a square. The class structure of the modular ring Z4 can be used to illustrate the result for various Pythagorean triples.

Five Smarandache conjectures on primes
Original research paper. Pages 119—120
M. Perez
Full paper (PDF, 46 Kb)

Strong Bertrband’s postulate revisited
Original research paper. Pages 121—123
L. Panaitopol
Full paper (PDF, 92 Kb)

On the 125-th Smarandache’s problem
Original research paper. Pages 124
K. Atanassov
Full paper (PDF, 37 Kb)


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

Comments are closed.