We offer surprises about     

in our e-book       Prime Passages to Paradise

by H. PeterAleff




Footnotes :




5 As quoted in George F. Simmons: "Calculus Gems -- Brief Lives and Memorable Mathematics", McGraw-Hill, New York, 1992, page 198 middle.




6 Amy Adams: "The Triumph of Hipparchos: A near-dud satellite helps solve a cosmic conundrum", Astronomy, December 1997, pages 60 to 63. Contrary to what the title implies, the author reported that the discrepancy between these ages was then reduced but still not fully resolved.




7 See Brian C. Chaboyer: "Rip Van Twinkle: The oldest stars have been growing younger", Scientific American, May 2001, pages 44 to 53. The more recent research described here claims the astronomers’ star age estimates were too high, and the universe obeys logic again.




8 As reported by Ian Stewart in "Mathematical Recreations -- Feedback", Scientific American, December 1997, page 121 left.





9 John H. Conway and Richard K. Guy: "The Book of Numbers", Copernicus - Springer Verlag, New York, 1996, pages 146 and 220.





10 George F. Simmons: "Calculus Gems ...", McGraw-Hill, New York, 1992, page 103 bottom.




11 Euler’s initial version was n2 - n + 41, and Legendre added the plus sign in 1798; both equations produce the same string of primes, and the latter version has become known as Euler’s polynomial. See R.A. Mollin: "Prime- Producing Quadratics", The American Mathematical Monthly, June-July 1997, page 529.





12 Paulo Ribenboim: "The New Book of Prime Number Records", Springer- Verlag, New York, 1996, pages 199 and 200. We will also compare this formula farther below with  various other prime- producing equations.





13 Using a 15-billion light year diameter (about 1.5 · 1028 cm) and the "classical" or pre- relativistic electron diameter of 2.8 x 10-13 cm, as given in Rita G. Lerner and George L. Trigg: "Encyclopedia of Physics", Second Edition, VCH Publishers, New York, 1991, page 290 right, top.





14 Neil Gershenfeld and Isaac L. Chuang: "Quantum Computing with Molecules", Scientific American, June 1998, pages 66 to 71, see page 66.





15 Announced on December 7, 2001, in The Prime Pages.  It was found by Michael Cameron with the Prime 95 prime- hunting program by George Woltman at the GIMPS site.




16 For earlier entries, see Paul J. Campbell, reviews editor: "GIMPS discovers 37th known Mersenne prime", Mathematics Magazine, April 1998, page 152. See also Keith Devlin: "World’s Largest Prime", Focus, the Newsletter of the Mathematical Association of America, December 1997, page 1.








  Volume 1: Patterns of prime distribution


in "polygonal - number pyramids"    


You are on page

Prime Facts

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

1.3. Laws and Order in Primeland

The square sieve of Eratosthenes hides this order inherent in the number line so well that many of the mathematicians who followed him have described primes as a disorderly- appearing bunch. For instance, the prolific and immensely influential master mathematician Leonhard Euler (1707-1783) expressed in 1751 his bafflement about the impenetrability of the primeland thicket:

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."5

Since primes are the basic building blocks of the number universe from which all the other natural numbers are composed, each in its own unique combination, the perceived lack of order among them looked like a perplexing discrepancy in the otherwise so rigorously organized structure of the mathematical world.

The apparent contradiction matches the head- scratching factor of the recent conundrum in astronomical research which said that our physical universe appears to be younger than its oldest stars6: How can so much of the formal and systematic edifice of mathematics, the science of pattern and rule and order per se, rest on such a patternless, unruly, and disorderly foundation?

Or how can numbers regulate so many aspects of our physical world and let us predict some of them when they themselves are so unpredictable and appear to be governed by nothing but chance?

Fortunately for people with scratch- weary scalps, the astronomical controversy has meanwhile been resolved7, and number investigators are also well aware that the ancient legend of utter chaos among the primes is not really true. They have described many patches of order in their midst.

Most recently, some researchers went as far as to compare groups of prime numbers in binary notation with biological sequences of DNA and RNA, and they were able to deduce by analogy that primes are not distributed at random but "that there are patterns, of some kind, in their sequence"8.

1.3.1.  Numerical versus physical universes

Already long before such novel approaches hinted that order must reign in primeland, researchers learned about the existence of laws there, roughly in parallel with their figuring out the laws that govern the physical universe.

A few decades after the mathematician and astronomer Johannes Kepler (1571- 1630) found in 1610 the rules behind planetary motion, the eminent amateur adept Pierre de Fermat (1601- 1665) discovered that all prime numbers of the form 4n + 1 are sums of two squares9

In 1640, Fermat also proposed his "little theorem", a fundamental rule about the remainders from numbers raised to prime and prime minus one power and then divided by that prime.

And about half a century after Isaac Newton (1642 to 1727) showed that Kepler’s empirically derived rules were mathematical necessities and physical laws, Euler published in 1736 that "little theorem’s" first proof 10 and so confirmed that primes are as law-abiding as planets.

With Newton’s laws as their guide, the astronomers began to understand the workings of the solar system to the point of predicting the appearance of comets. Edmond Halley’s (1656 to 1742) comet returned in 1758 on its computed schedule as the first of its kind ever to do so.

And with his growing knowledge of the number world, Euler himself, despite the despair he had voiced earlier about ever finding some order among the primes, discovered in 1772 the surprising formula n2 ñ n + 41 which produces11 a comet- like streak of primes, uninterrupted for its first 40 entries and then continuing to yield a prime-dense tail that stretches as far as anyone has checked its results up to now12.

Five years later, Euler spotted several other prime-producing values for the constant in that formula. Since then, his successors have added more such formulae of the same and similar types and discovered various rules about the behavior of primes, to the point where they now know quite a few chapters of primeland’s lawbooks. They have also compiled a directory to the addresses of its accessible inhabitants and devised reliable ways to find some of those that dwell beyond the examined area.  To do so, they used powerful mathematical concepts and ever faster computing tools that let them survey or spot- check regions of counting far beyond even those with which their colleagues in astronomy stretch our minds.

To appreciate the distance and size of the prime-hunters’ trophies, consider, for instance, that our entire known universe is less than 1042 electron diameters13 across, give or take a few billion light-years. It could thus not contain more than at most 10126 of these smallest particles that we can still describe, at least in part, as matter.

Even these itsy- bitsy electrons would fit into that universe only if they were packed cheek to cheek throughout its vast and now virtually void space, more densely even than in the gravity- crunched interior of a neutron star where matter gets so compressed that an object the size of a skyscraper on Earth would fit into a thimble.

And even if we tried to pack that space with still smaller particles from the quantum realm, or with their yet harder- to- grasp components, and even if tomorrow’s surprise discovery of a consistently repeated multi- decimal error in all the redshift calculations ever performed suddenly made the universe thousands or millions of times older and bigger than before that Big Oops, none of these changes would greatly affect the basic argument.

No matter how we massage the data, it would still take, at best, not much more than a couple hundred digits to encompass all the conceivable matter within our horizon and all the space it could possibly occupy.

Time does not get us much farther: computer experts estimate that finding the prime factors of a number with 400 digits would take the currently fastest supercomputers several billion years14. Toss in a handful more digits, and finishing the task with the same brute- force method would take even the next generations of hardware longer than all the world’s conceivable past.

Then add the digits from your other hand and your toes, and by the time those computers finish their factoring, they and we will all have evaporated in tiny quantum jumps from the black hole that will have gobbled our world eons before that far far future. 

(Let us just hope that future astronomers can thin down the hidden matter which the current ones postulate to doom our descendants to that Big Crunch).

Now compare those few- hundred- digit quantities from near the upper limits of our cosmological conceptions with the world’s largest known prime that had  4,053,946 digits15, as of December 2001, and keeps getting more: thousands of computers are churning in the search for still larger primes, and for the per-digit cash prizes promised to their finders16. The chances are therefore good that even this amazing number may be surpassed by the time you read this.




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