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Prime number alignments

long forgotten but long ago vital

An exploration of patterns in the distribution of primes, and some evidence for their ancient perception

The first volume in this series discusses different ways of organizing the natural numbers to make them reveal some of the order in the distribution of primes.  Primes are numbers which have no factors except one and themselves, and they are the basic building blocks from which all other numbers are made

One of these organizing arrangements allows a simple proof for the long- standing twin prime conjecture that the quantity of twin primes, or pairs of primes which are separated only by the even entry between them, is infinite.

You will further find here descriptions and illustrations of some surprising patterns formed by primes

These visually striking patterns appear already in the first few thousand entries when you write the consecutive natural numbers in simple triangle- shaped arrays where the ends of the successive layers are, for instance, the successive squares 1, 4, 9, 16, 25, and so on, and you then mark the primes.

Those patterns are densest near the apex of those triangles but continue to higher regions in gradually more diluted form.  The beginnings of the first three are drawn on the array- posters which show the structure of the number world the way the Periodic Table of Chemical Elements shows that of the physical world.

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Volume two shows that some of these easily generated arrays, which require no advanced mathematical knowledge, appear to have been of great interest to the ancient Egyptian pyramid builders.

A comparison of the most prominent alignments of primes in those triangle- shaped number- lists with the passages into (or out of) those triangular- cross- section stone heaps suggests that their architects may have included prime number stairways among the various magical ways to assure the ascent of their dead kings to heaven.

The similarities imply that some of the discoveries ascribed to the much later Greek mathematician Pythagoras, and more, may already have been known to the Old Kingdom Egyptians.

They also highlight the underappreciated but vital role of numbers in the pharaonic religion where the pantheon heads typically ruled through Thoth the god of number and of magic who always kept his central job as their messenger and executive vizier while his remote nominal bosses came and went.

The originators and followers of that number- magic doctrine seem to have seen the world of those invisible and intangible but powerful and timeless entities as a bridge to the equally abstract domain of the gods.

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The illustrations for Volume 1 and the posters are presented as .gif raster format files, linked to the blue- framed thumbnail pictures so that you can view them with any picture- enabled browser and without plug- ins.  Unfortunately, to keep the file sizes and load times down, the resolution of those raster pictures is limited and does not let you appreciate details.

However, the view vector version links near those thumbnails offer you the option to examine the same illustrations in the more flexible Drawing Web Format. The file sizes of the drawings posted here in this vector format are even larger because they contain much more detail, but the pictures remain sharp at any magnification and allow you to zoom in up close to the areas that interest you.

To see or print these .dwf format vector drawings, you need a vector viewer program.  An easy-to-use example is Autodesk Design Review which you can download free from

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Unless you have a wall- size screen, even the vector display makes you choose between overall view and close-up.  To see at the same time the whole and the part, it is better to print the drawings on paper, particularly the posters.  The typical example is meant to be printed on standard 8½ x 11 inch paper, except for the posters which are designed for printout on four to six "legal- size" sheets (8½ x 14 inches) which can then be taped together.

Continue to the Table of Content for Volume 1.

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