by Nick Mee

Issue 25

Submitted by plusadmin on April 30, 2003

in

* 25

* Arthur C Clarke

* cryptography

* Fundamental theorem of arithmetic

* number theory

* prime number

* prime number distribution

* prime number spiral

* Riemann hypothesis

* Riemann zeta function

* zeta function

May 2003

Jeserac sat motionless within awhirlpool of numbers. The first thousand primes, expressed in the binary scale that had been used for all arithmetical operations since electronic computers were invented, marched in order before him. Endless ranks of 1's and 0's paraded past, bringing before Jeserac's eyes the complete sequences of all those numbers that possessed no factors except themselves and unity. There was a mystery aboutthe primes that had always fascinated Man, and they held his imagination still.

Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was hispleasure to hack his way through the arithmetical jungle and sometimes he discovered wonders that more skilful explorers had missed.

He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh. Jeserac had done this a hundred times before and it had never taught him anything. But he wasfascinated by the way in which the numbers he was studying were scattered, apparently according to no laws, across the spectrum of the integers. He knew the laws of distribution that had already been discovered, but always hoped to discover more.

from The City and the Stars by Arthur C. Clarke (1956)

The building blocks of arithmetic

Carl Friedrich Gauss

In the words of the great Germanmathematician Carl Friedrich Gauss: "Mathematics is the Queen of the Sciences and Arithmetic is the Queen of Mathematics." The modern name for the branch of mathematics that Gauss was referring to as Arithmetic is Number Theory - the study of the properties of the positive whole numbers or integers. The 19th century mathematician Kronecker famously claimed that "God made the integers, all the rest is thework of man."

The fundamental building blocks of Number Theory are the primes. These are the numbers: 2, 3, 5, 7, 11, 13,... defined as the whole numbers that cannot be divided exactly by any other whole number, excluding the trivial division by the number 1. Primes cannot be broken down into simpler components; they play a role in mathematics that is similar to the role of the elements inchemistry. From the 100 or so chemical elements it is possible to synthesize the millions of compounds that are studied by chemists. The Fundamental Theorem of Arithmetic, which was proved by Euclid, states that

All positive whole numbers are either primes or they can be uniquely decomposed into a product of primes.

For instance:

| | | | | |

| | | | | |

| | | | | |

|| | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

If we take all the primes less than 300, we find that there are just 62 of them:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,

31, 37, 41, 43, 47, 53, 59, 61, 67, 71

73, 79, 83, 89, 97, 101, 103, 107, 109, 113,

127, 131, 137, 139, 149, 151, 157, 163, 167, 173,

179, 181, 191, 193, 197, 199,211, 223, 227, 229,

233, 239, 241, 251, 257, 263, 269, 271, 277, 281,

283, 293.

25 of these primes are below 100, 21 are between 100 and 200 and 16 are between 200 and 300. It looks as though the primes become more spread out as their size increases. If we look further we find that between 10,000 and 10,100 there are just 11 primes, between 100,000 and 100,100 there are just 6. This seems to...

## Поделиться рефератом

Расскажи своим однокурсникам об этом материале и вообще о СкачатьРеферат