Dr John C Turner

Hi again; second blog of the day!

Just as I was working to finish off CNB4 this morning, I got a grim message, on the screen: my PC couldn’t find the url of my website. Oh dear, thought I!  My son tells me not to worry; and I have already got a grandson-in-law working on the problem. I have faith that they will soon pull it back into the fold.

But that doesn’t stop me from editing a new Blog post,  so I will prepare one giving a few of my thoughts and feelings about the objects which I have called ‘cycle-numbers’. And put down a few comments on why I think they are worth studying.

First let me ask you the old, old, old philosophical question:  “What is Life”.

A good answer occurred to me after I discovered that my website was now only a blank on the screen:      ” A mathematician’s life is one darned problem after another.”

And that is exactly what you will find if you try to prise number secrets out of the (0,1)-patterns you find in T and C: one darned problem after another. And I do hope you are looking, and trying. And that whenever you find one that interests you, that you will try to find relationships with other patterns that you have spotted. Then there are all manner of things that you can try to do; I suppose you might judge that the prize things to obtain is new knowledge about the primes and their distribution amongst the natural numbers. But there is plenty to be said about each of the natural numbers, in general, before trying to discover how the primes arise, and what their roles are in relation to the non-primes (composites). I have indicated a few of such things in earlier blogs; here I just want to say a little about some of the most striking things, which seem to me to lie at the very bottom of the natural number system, from the point of view of my cycle-number representation of it.

(i) It fascinates me that once the alphabet {0,1} is given, and the zero number (0,1)000 … is defined, all that is needed to produce the cycle-number triangle T is to apply the double-cycling method, using the neck-ties, to obtain all the fundamental cycles of the natural numbers in the rows of T.

(ii) I think (poor prejudiced me) that this development of the number system is more basic [just cycling of (0,1)-patterns] than the one obtained from Peano’s axioms. He starts with axiom (1), stating baldly that zero is a number, which contains two undefined terms; then a further four axioms establish an infinite sequence of successors of zero, say {S(0), SS(0), SSS(0), …}, which after lengthy discussion and logical analysis and further definitions, becomes identified with our familiar sequence {1, 2, 3, , …}. And we arrrange the elements of this in a row of gleaming dots in our mind, along our mental number line, which some neurologists now tell us we are born with, to use for comparing group sizes (e.g. groups of lions attacking our tribe!)

(iii) Whereas,  each of my cycle-numbers emerges from its chrysalis in T (or the matrix C), and immediately takes off like a butterfly. fluttering along an infinite line of its own, its (0,1)-pattern growing steadily, in cycles, to infinity. One can imagine it as a complete, though infinite, sequence (I can, anyway!). And I see it also, as carrying a long trail of information for me, about the whole set of natural numbers N. A bit like the early Tiger Moth planes, which flew trailing a long ribbon or banner, advertising some event or other. Not only is the cycle-number in constant motion, pedalling along its own line (i.e. row) in C, but when we look back and forth in its row, we see that it has a past, a present, and a future.