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

**Associated Legendre polynomials and Morgan—Voyce polynomials**

*Original research paper. Pages 125—134*

A. F. Horadam

Full paper (PDF, 342 Kb)

**One extremal problem. 9**

*Original research paper. Pages 135—137*

K. Atanassov

Full paper (PDF, 81 Kb)

**Number theoretic aspects of a combinatorial function**

*Original research paper. Pages 138—150*

L. Halbeisen and N. Hungerbühler

Full paper (PDF, 434 Kb) | Abstract

We investigate number theoretic aspects of the integer sequence A000522 of Sloane’s On-Line Encyclopedia of Integer Sequences. This integer sequence counts the number of sequences without repetition one can build with n distinct objects. By introducing the notion of the “shadow” of an integer function – which is related to its divisors – we treat some number theoretic properties

of this combinatorial function and investigate the related “irregular prime

numbers”.

of this combinatorial function and investigate the related “irregular prime

numbers”.

**The Cardano family of equations**

*Original research paper. Pages 151—162*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 391 Kb) | Abstract

**#!**The polynomial expansion of the Diophantine equation xn = (x — p)n -f (x — g)n, p, g € Z + ,n > 2, yields roots of the form ( ( p + ? ) + y) where yis a non-integer zero of a Cardano cubic polynomial of the form y3 — 6pqy — 3pq(p + g). This is a corollary to Fermat’s Last Theorem. The characteristics of this family are illustrated for n = 3 ,4 , . . ., 9. For nodd, y0can be represented by (n – l)(2pg + e)J ,and for n even there arc two real values of y0, (n – l)(2pq + e)5 and — (2pg + d) 3, where d, earc real non-integer parameters. For a given n, e is simply related to p / q ,p < g, and to a parameter E which is linear in n. The corresponding curves indicate the non-integral nature of y, n > 2.