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by H. PeterAleff



Footnotes :




17 Trinh Xuan Thuan: "The Secret melody - And Man Created The Universe", Oxford University Press, 1995, pages 119 and 230/231.




18 Oystein Ore: "Number Theory and its History", McGraw-Hill, New York, 1948, page 71 bottom; see also Euclid: "The Elements", Book IX, Proposition 36, Dover, New York, 1956, Vol. 2, pages 421 to 426.




19 George F. Simmons: "Calculus Gems -- Brief Lives and Memorable Mathematics", McGraw-Hill, New York, 1992, page 278 top.




20 Lee Smolin: "The Life of the Cosmos", Oxford University Press, New York, 1997, as reviewed in Astronomy, March 1998, pages 100 and 102. See also George Musser: "Inconstant Constants", Scientific American, November 1998, pages 24 and 28.




21 Calvin C. Clawson: "Mathematical Mysteries -- The Beauty and Magic of Numbers", Plenum Press, New York, 1996, chapter on "Deepest Mysteries", see page 270. For a list of the errors remaining in prime quantity estimates with various methods up to 1016, see John H. Conway and Richard K. Guy: "The Book of Numbers", Springer Verlag, New York, 1996, page 146.

Riemannís "refinement" is only 327,052 primes off from the actual count, or about 0.0000001%, and his unproven "hypothesis" would come still closer.




22 Paulo Ribemboim: "The New Book of Prime Number Records", Springer Verlag, New York, 1996, page 241.




23 "Focus: The Newsletter of the Mathematical Association of America, August / September 2000, page 3. For details, see the website of the Clay Mathematics Institute at




24 Phillip A. Griffiths: "Mathematics at the turn of the Millennium", in The American Mathematical Monthly, January 2000, pages 1 to 14, page 9 middle. For details, check directly the website of the Clay Mathematics Institute.




25 George F. Simmons: "Calculus Gems -- Brief Lives and Memorable Mathematics", McGraw- Hill, New York, 1992, page 205.








  Volume 1: Patterns of prime distribution


in polygonal - number "pyramids"    


You are on page

Prime Problems

  0  1   2   3   4   5   6   7   8     + 10   11  12  13  14  15  16  17  18

1.3.2.  The ongoing search for prime laws

Regardless how much greater the number world may be than anything we can imagine, its laws apply to all its regions (with a few exceptions for its founding units up to 3 or 4, just as the physical laws for the starry world may not all apply to the first 10-43 seconds of the Big Bang17).

The prime-hunters discovered the above record holder prime with the same formula that had produced most of the runner-ups, and that allowed them to prove these numbers are prime although they far exceed anyoneís ability to find their factors if they had any.

That formula was known since antiquity18 for its association with so-called perfect numbers.  These equal the sum of their factors and thereby greatly impressed number mystics of the Pythagorean persuasion. 

The primes obtained with that formula are currently named after the monk Marin Mersenne (1588-1648) who discussed its link with primes.

The participants in the prime race are confident they can find more such primes because they know that even their biggest catches, immense beyond any imaginable human grasp, are infinitesimally tiny when compared with the much bigger primes that lay sprinkled along the number line into infinity.

And mathematicians have proved that even those numbers far, far, far beyond their reach still all obey the same laws -- a feat no other lawmakers have ever achieved.

Number theorists have shown that primes obey statistical laws at least as stringent as those of the quantum-level equations for the physical world, and they hope to refine the resolution of their methods to the point of discerning a pattern.

Beginning with the mathematical giant Carl Friedrich Gauss (1777-1855), they first measured the density of primes in the accessible parts of the number world and then proved the prime number theorem which says that the quantity of primes up to any number x tends, as x gets larger, towards the ratio of that x divided by its natural logarithm.

Some writers call this simple law one of the most remarkable facts in all of mathematics19.  It allows us to know the large- scale density of primes even in regions where no one can count them, and that puts the mathematicians ahead of the astronomers who still have to reconcile their greatly differing estimates for the density of matter in the known physical universe.

Similarly, while physicists have no idea to what degree the physical laws can remain accurate in extreme circumstances such as below the subatomic level, or inside black holes20, mathematicians have narrowed down the size of the error in the approximations their prime number theorem supplies, and they are hard at work trying to reduce it further.

Their current focus is to prove or disprove a possibly even better formula which Bernhard Riemann (1826 to 1866) proposed in 1859.  He was the mathematical genius who came up with the curved geometry Einstein would use later to construct and describe his universe, and he developed intricate analytic methods to improve on that theorem. 

If Riemann's function could be proven, it would nail down the quantity of primes up to any large number so accurately that it should allow to locate the primes anywhere within the number sequence and so to show that they follow a very subtle and precise pattern21.

As of 1986, a team of researchers had verified, with over a thousand hours on a supercomputer, that Riemannís guess holds true for the first billion and a half values obtained with his function22.

Meanwhile, in November 2001, Sebastian Wedeniwski at IBM Germany announced that even longer calculations, distributed over many thousands of desktop computers, have extended that verification to over ten billion entries, and he posted details at

However, this sampling says nothing about the rest of the sequence. A single exception anywhere would show the hypothesis to be false, so proving it for all numbers remains one of the foremost mathematical research challenges for the 21st century. 

To underscore its great importance, the Riemann Hypothesis is now one of the seven "Millennium Prize Problems" for each of which the Clay Mathematics Institute offers a prize of a million dollars23.

Experts in the field call the verification of this hypothesis "perhaps the deepest existing problem in pure mathematics"24 and "the most important unsolved problem of mathematics, probably the most difficult problem that the human mind has ever conceived"25.

The climb to this highest and innermost bastion of the numberworld maze may be steep and thorny, but the view from its summit is expected to allow the intrepid climbers, or their future disciples, to deduce the design of that otherwise apparently impenetrable labyrinth.




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