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No. 42: Nov-Dec 1985

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The Fabric Of Prime Number Distribution

Mathematicians and many non-mathematicians have soft spots in their hearts for numbers that cannot be subdivided; that is, the prime numbers. No one has ever been able to figure out any foolproof system to their occurrence or how to generate them all by formula. Some primes, such as 11 and 13, 17 and 19, etc., come in pairs; but most don't. And some formulas are particularly good at generating primes, but they all fail somewhere. One such formula was discovered by Leonhard Euler, the great mathematician: n2 + n + 41 This formula works for n = 0, 1, 2,...39; but fails at n = 40.

An interesting consequence of Euler's formula can be made apparent when all numbers from 41 to 440 are written in a square spiral, like so:

Euler's formula as a square spiral

All of the numbers on the diagonal indicated are Euler formula primes, even when the spiral is expanded to 20 x 20. However, when the 20 x 20 spiral is examined closely, many of the other primes -- those not generated by Euler's formula -- also tend to line up on diagonals. This is a most intriguing characteristic, one which goes far beyond the 20 x 20 array mentioned above. The computer-generated display shown below lays out a huge square spiral, with each prime a bright dot. The picture has a pronounced diagonal texture. Why this is so is a mathematical mystery. (Crypton, Dr.; "Prime Numbers and National Security, " Science Digest, 93: 86, October 1985. ) (Does the diagonal fabric of primes have any practical significance ? At the moment no one knows. Again and again, abstruse mathematical structures have turned out to mirror phenomena in the world we call "real". WRC)

Computer generated display of prime numbers as bright dots

From Science Frontiers #42, NOV-DEC 1985. 1985-2000 William R. Corliss

Science Frontiers Sourcebook Project Reviewed in:


  • "A sourcebook of unexplained phenomena is therefore a valuable addition to a collection of scientific literature. William R. Corliss has provided this in the past with his source books of scientific anomalies in several subjects, and now he has provided it for astronomy. He has done an excellent job of collecting and editing a large amount of material, taken in part from scientific journals and in part from scientific reporting in the popular or semi-scientific press." -- "The Mysterious Universe: A Handbook of Astronomical Anomalies", reviwed by Thomas Gold, Cornell University, in Icarus, vol.41, 1980

  • "An interesting, systematic presentation of unusual weather [..] This book is recommended for a general audience" --"Corliss, William R., Tornados, Dark Days, Anomalous Precipitation, and Related Weather Phenomena, Sourcebook Project, 1983.", revieweed in Choice, September 1983
  • "..the science is necessarily somewhat speculative, but Corliss's symthesis is based on reputable sources." -- "Corliss, William R. (Compiler). Lightning, Auroras, Nocturnal Lights, and Related Luminous Phenomena" reviwed by Joseph M. Moran, Univ. of Wisconsin in Science Books and Films, Sep/Oct 1983

  • "Before opening the book, I set certain standards that a volume which treads into dangerous grounds grounds like this must meet. The author scrupulously met, or even exceeded those standards. Each phenomenon is exhaustively documented, with references to scientific journals [..] and extensive quotations" -- "Book Review: The moon and planets: a catalog of astronomical anomalies", The Sourcebook Project, 1985., Corliss, W. R., Journal of the Royal Astronomical Society of Canada, Vol. 81, no. 1 (1987), p. 24., 02/1987