Volume 4, 1998, Number 3

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

Nagell’s totient revisited
Original research paper. Pages 93–100
Pentti Haukkanen and R. Sivaramakrishnan
Full paper (PDF, 320 Kb) | Abstract

Nagell’s totient θ(n, r) counts the number of solutions of the congruence (*) n = x + y (mod r ) under the restriction (x, r) = (y, r) = 1. In this paper we evaluate the number θ(n, r, q) of solutions of the congruence (*) under the restriction (x,r) = (y,r) = q, where q|r, via Ramanathan’s approach to class-division of integers (mod r).

On the 40-th and 41-st Smarandache’s problems
Original research paper. Pages 101–104
Valentina V. Radeva and Krassimir T. Atanassov
Full paper (PDF, 143 Kb)

On some arithmetic functions
Original research paper. Pages 105–107
Mladen V. Vassilev-Missana and Krassimir T. Atanassov
Full paper (PDF, 100 Kb)

On sums of pairs of squares and cubes
Original research paper. Pages 108–112
J. H. Clarke, A. G. Shannon and J. V. Leyendekkers
Full paper (PDF, 178 Kb)

Recurrence relation analysis of Pythagorean triple patterns
Original research paper. Pages 113–122
J. M. Rybak, J. Leyendekkers and A. G. Shannon
Full paper (PDF, 306 Kb) | Abstract

This paper explores recurrence relations in their role of providing internal generators of Pythagorean triples. While the relation of Pellian recurrence relations to diophantine equations in general is not new, this paper classifies the internal generators according to their parity and primality.

First-order recurrence relations for the Chebyshev polynomials and associated function
Original research paper. Pages 123–128
Richard L. Ollerton and Richard N. Whitaker
Full paper (PDF, 3562 Kb) | Abstract

The Chebyshev polynomials of the first kind, Tn(x) = cos(n cos−1 x) (n integer, |x| < 1), satisfy the second-order recurrence relation Tn + 2 = 2xTn+1 − Tn, T0 = 1, T1 = x. It is shown that they also satisfy the first-order recurrence relation Tn+1 = xTn + r((1 − x2)(1 − Tn2)), T0 = 1, where the function r is defined by r(p(x)2) = slc(p(x))p(x) for polynomial p(x) and slc(p(x)) denotes the sign of the leading coefficient of p(x).

Associated Chebyshev polynomials, satisfying Xn+2 = 2a(x)Xn+1 − Xn, X0 = x0, X1  polynomial, for polynomial a(x), are then defined and the corresponding first-order relation given. An example of non-polynomial a(x) leading to the functions Vn(x) = sin(n sin−1 x) is also mentioned together with a more general first-order recurrence relation for the non-polynomial case.

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