Problems & Puzzles: Puzzles
Puzzle 274. Sierpinski triangles from prime numbers
This puzzle is a charming. It joins the beauty and the beast, that is to say the Sierpinski triangles (ST from now on) and the Prime numbers.
Please open you mind and see this issue proposed by James Thomas, from London.
Well, Thomas has found how to construct the apparently complex to draw ST (a fractal) by a very simple and unexpected way. I invite you to do it by your self in your own Excel worksheet .
Amazing!... isn't it?
In his first two emails (June 17 & 18, 2004), Thomas wrote:
By the very simple trick of making a 25% zoom of the entire worksheet , it happens that the zero values became invisible (++), making easier to see just the ST. Formatting the column width to 2, you obtain triangles with legs approximately of the same length.
At this point I remembered that somebody else practiced similar handling of the prime numbers, that is to say, calculated consecutive absolute differences of adjacent numbers, starting with the prime numbers. This was exactly the matter of the so-called Gilbreath's Conjecture (GC from now on)
I asked Thomas if he was aware of any connection of his issue with this conjecture and recommended him to read at least the Eric Weisstein's article about the matter.
His response was "I had not heard of this. I will read it with interest". Later he added "I have read Gilbreath's Conjecture and see that it is equivalent to the top left to bottom right diagonal in the Prime Graph I sent you, (2,1,1,...)."
As a matter of fact the Gilbreath's domain of calculations is the lower-left corner of the Thomas matrix, while the ST domain is the upper-right corner; the diagonal mentioned is the intersection of both domains and is the specific conjectural affirmation of the Gilbreath's conjecture (GC).
For sure, many readers already know that A. M. Odlyzko "has checked it [the GC] for primes up to π(1013)≈3.1011" (**).
Regarding the specific issue of the GC "Hallard Croft and others have suggested that it has nothing to do with primes, but will be true for any sequence of 2 and odd numbers, which doesn't increases too fast, or have too large gaps. Odlyzko discuss this [Math. Comput. 61(1993)373-380]"
Thomas has also examined the persistence of the ST in the upper-right corner of the matrix, using several sequences other than the prime numbers, and in short he knows that the ST remain in the following cases: substituting in the first column the prime numbers sequence for:
a) Natural numbers
= ai+1; a1=1)
So, probably - and this is the way I think is better said - the ST, as obtained by Thomas, are related to certain class of sequences to which the prime numbers sequence belongs.
But there is still another possibility that I have devised while preparing the blueprint for exposing the Thomas procedures.
You may also produce the ST in the upper-right corner of the matrix by the following way, independent of any specific sequence fed in the first column:
Create a matrix such that:
Maybe this is the real object found by Thomas: a very simple matrix-mechanism in order to build arithmetically the beautiful ST fractal.
But this very simple matrix-mechanism, in the way I have exposed it, is now independent of any sequence fed to the first column.
In this sense maybe the important subsidiary question is to find out why the prime sequence belongs to the class of sequences that produces an upper-left to bottom-right diagonal with a constant value (question that could have been already responded by Odlyzko and others in the article previously quoted)
Before pose our own questions, I really encourage to you, to construct your own Excel worksheet and play with it - changing the type of sequences fed to the first column, and changing the parameter of the particular sequence you want to visualize. The visual spectacle is really enjoying and pleasant. Maybe you can find something new, over sighted by Thomas and me; maybe you can find some other interesting graphs (fractal or not) changing the fed function... who knows?
1. Is the Thomas procedure for getting the ST an original procedure, or do you know a previous exposition of it?
2. Is the Thomas procedure of any help in order to demonstrate the GC?
(++) You can get similar result from the [excel] menu. Select Tools, Options, select the View tag, and un-tick the field zero values. This will hide zero values irrespective of the zoom of the worksheet.
(**) Cfr. A10, pp. 25-16, R. K. Guy, UPiNT.