A representation of the natural numbers by means of cycle-numbers, with consequences in number theory
- Publication date
- Publisher
Abstract
In this paper we give rules for creating a number triangle T in a manner
analogous to that for producing Pascal’s arithmetic triangle; but all of its
elements belong to {0, 1}, and cycling of its rows is involved in the creation.
The method of construction of any one row of T from its preceding rows will
be defined, and that, together with starting and boundary conditions, will
suffice to define the whole triangle, by sequential continuation.
We shall use this triangle in order to define the so-called cycle-numbers,
which can be mapped to the natural numbers. T will be called the ‘cyclenumber
triangle’.
First we shall give some theorems about relationships between the cyclenumbers
and the natural numbers, and discuss the cycling of patterns within
the triangle’s rows and diagonals. We then begin a study of figures (i.e. (0,1)patterns,
found on lines, triangles and squares, etc.) within T. In particular,
we shall seek relationships which tell us something about the prime numbers.
For our later studies, we turn the triangle onto its side and work with a
doubly-infinite matrix C.
We shall find that a great deal of cycling of figures occurs within T and
C, and we exploit this fact whenever we can. The phenomenon of cycling
patterns leads us to muse upon a ‘music of the integers’, indeed a ‘symphony
of the integers’, being played out on the cycle-number triangle or on C. Like
Pythagoras and his ‘music of the spheres’, we may well be the only persons
capable of hearing it!
Keywords: cycle-number triangle, cycle-number, prime cycle-number