Volume 11, 2005, Number 4

Volume 11Number 1Number 2Number 3 ▷ Number 4

On the structure of certain counting polynomials
Original research paper. Pages 1–11
R. Valenza
Full paper (PDF, 135 Kb) | Abstract

We consider a natural generalization ζ(k)(s) = Σαn/ns (k ≥ 2) of the Riemann zeta function that arises from a modification of its classical Euler product expansion, for the most part here concentrating on the case k = 2. The associated coefficients αn correspond to a counting problem that may be addressed via a family of multivariable generating functions. Examples computed via symbolic manipulation suggest a recursive structure for these functions, which we prove. With this result in hand, the calculation of the αn may be facilitated by a more efficient, double modular algorithm, as worked out in a detailed example. We conclude with some observations and questions for the case k > 2.

On two new 2-Fibonacci sequences
Original research paper. Pages 12–16
Krassimir T. Atanassov
Full paper (PDF, 1604 Kb)

Fermat and Mersenne numbers
Original research paper. Pages 17–24
J. V. Leyendekkers and A. G. Shannon
Full paper (PDF, 114 Kb) | Abstract

Fermat numbers (Fn = 22n + 1) and Mersenne numbers (Mm = 2m − 1), m odd, are compared on the basis of integer structure, using the modular rings Z4 and Z6. The two numbers fall in different classes and this results in different composite row structures and different potentials for the formation of primes. The constraints on 2n and the right end digits for Fn result in fewer numbers over a given range than those for Mm. This is shown with two functions, which link the two numbers and show that Fn = (2x2 + y2 − 1 + 1): for primes y = n, but when n > 4, yn.

On the sum of equal powers of the first n terms of an arbitrary arithmetic progression. II
Original research paper. Pages 25–28
Peter Vassilev and Mladen Vassilev-Missana
Full paper (PDF, 1146 Kb)

Volume 11Number 1Number 2Number 3 ▷ Number 4

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