Volume 3, 1997, Number 3

Studying the unique minimal and maximal integer Zeckendorf representations by Pell numbers [2], [3], [4], [5] led to the consideration [6, eqn. (2.7)] of those numbers which are common to both representations, namely, the MinMax numbers. This idea was then carried over to Jacobsthal numbers [7, eqn. (3.1)]. (Earlier, minimal and maximal representations by Fibonacci and Lucas numbers had been investigated in [1].)Here, the corresponding situation existing for Morgan-Voyce numbers is to be disclosed. Though the results are perhaps not quite so elegant as those for Pell numbers, they are nevertheless of intrinsic interest and value.

The essential characteristics of integers and the relationships with their powers are explored within the framework of the modular ring ℤ₄ in order to analyse why odd powered triples with exponents greater than unity cannot exist in integer form. Two methods are given which exploit old expansion and reduction techniques in a new way. By way of conclusion the second method is also illustrated by reference to Pythagorean triples.

We generalize von Mangoldt’s function and certain arithmetical products of trigonometrical functions and Euler’s gamma, function in terms of Narkiewicz’s regular convolutions. We give arithmetic evaluations for these products and we establish asymptotic formulae for them in case of cross-convolutions, investigated by the first author in previous papers.

Difference equations are often convenient in the mathematical modelling of medical problems because many physiological properties, such as the measurement of plasma glucose levels, tend to be assessed at discrete time intervals. Ollerton and Shannon [4] describe one such linear difference equation in detail in the context of the development of an artificial beta-cell. A partial non-linear difference equation which arose in a similar context is given by

(1 + u_i,j+i² − 2u_i,ju_i,j+1 + u_i,j²)(u_i+2,j − 2u_i+1,j + u_i,j) − 2((u_i+1,j − u_i,j) u_i,j+1 − (u_i+l,j − u_i,j) u_i,j)
+ (l + u_i+1,j² − 2u_i+l,ju_i,j + u_i,j²)(u_i,j+2 − 2u_i,j+1 + u_i,j) = 0.

The purpose of this note it to convert this difference equation into the corresponding partial differential equation and to examine the resulting minimal surface. This is done formally without digressing into the issues associated with mapping from the integers to the reals.

Fermat used the method of infinite descent to show that the area of a Pythagorean triangle can never be a square. Here we use the modular ring ℤ₆ [1] to obtain the same result.