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.

I suppose that you already know what the ST are. If not please first go here:1, 2, 3.

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 .

• In the column A enter the prime numbers: 2 in A1, 3 in A2, 5 in A3, and so on ... . The more primes you enter the better for the graph intended. So I would recommend to enter at least 200 prime numbers in this column (1223 in the cell A200 *).
• In the first row, please enter just the prime 2. Please enter this prime number in at least 200 cells of the first row (from the cell A1 to the cell HZ1).
• Inside each cell other than the already entered before, enter a formula to calculate the absolute difference between the two adjacent integers in the adjacent-left column, as follow:

  (r, c) = abs[(r,c-1)-(r-1,c-1)] r=row #, c=column # .............(F1)

  In the Excel's typical syntaxes, for example, in the cell B2 enter the following formula '=abs(A2-A1)'. Then just copy it to the other cells.

• Ask for the entire worksheet for a column width of size 2.
• Ask for a 25% zoom view, and ... done it!

Amazing!... isn't it?

Just in case you haven't been able to get - for any reason - that precious image, I provide the jpg files prepared by James Thomas for this puzzle, on my request: 1, 2.

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 (a_i+1 = a_i+1; a₁=1)
b) Arithmetic progression (a_i+1 = a_i+b)
c) Geometric progression (a_i+1 = k.a_i, for k<=2; for any 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:

• Each cell of the first row has a constant value k>0
• Each cell of the upper-left to bottom-right diagonal has the same constant k value
• Each cell in the upper-right corner of the matrix - other than the already fed - is calculated by (F1)
• This matrix contains only 0 and k values. The ST are drawn by these cells whose value is k.

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?

Questions:

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?

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(*) For this purpose you will be greatly assisted by the Joe Crump's Excel addin.