Prime Numbers

Prime Numbers, An Odyssey

Prime numbers, a simple yet profound construct in arithmetic, have perplexed

generations of brilliant mathematicians. Its infinite existence was

known as early as Euclid over 2000 years ago, yet the prime

number theorem, regarding the distribution of prime numbers, was

only proven in 1896, following Bernhard Riemann’s study of its

connection to the Riemann zeta function. There remain still

numerous unsolved conjectures to this day.

The world records in prime numbers have been largely focused on

Mersenne prime 2p–1, named after French monk Marin Mersenne

(1588-1648), due to its long history and importance in number

theory, and the fact that modulo 2p–1 can be computed without

Bernhard Riemann

1826-1866

Pierre de Fermat

1601-1665

division for the efficient Lucas-Lehmer test. Currently the top 10

largest known primes are all Mersenne primes.

Two well-known types of prime pairs are, twin primes, where

both p and p+2 are primes, and Sophie Germain (1776-1831)

primes, where both p and 2p+1 are primes. Extending the

concept of Sophie Germain prime pairs, a chain of nearly

doubled primes is named after Allan Cunningham (1842-1928),

where Cunningham chain of the first kind has each prime one

more than the double of previous prime in chain, and where

Cunningham chain of the second kind has each prime one less than the double of

previous prime in chain. A variation of the form is known as bi-twin chain, that is, a

chain of twin primes where each twin pair basically doubles the previous twin pair.

Let’s look at some small examples to better understand these prime chains. 5 and 7 are

twin primes, 6 is their center. Let’s double 6, arriving at 12, whereas 11 and 13 are twin

primes again. So 5, 7, 11, 13 is a bi-twin chain of length 4, also known as bi-twin chain of

one link (a link from twin 5, 7 to twin 11, 13). The bi-twin chain can actually be split

from their centers, giving one Cunningham chain of first kind, and one Cunningham

chain of second kind. Now if we split through centers 6, 12 of bi-twin chain 5, 7, 11, 13,

those below the centers are 5, 11, a Cunningham chain of first kind, those above the

centers are 7, 13, a Cunningham chain of second kind. I call the first center, the number 6

in this example, the origin of the prime chain. From this origin you can keep doubling to

find your primes immediately adjacent to the center numbers.

There are also other prime formations known as prime constellations or tuplets, and

prime arithmetic progressions. Of interest to these prime pairs and formations, is that

their distribution seems to follow a similar but more rare pattern than the distribution of

prime numbers. Heuristic distribution formulas have been conjectured, however, none of

their infinite existence is proven (the twin prime conjecture being the most well known

among them [Goldston 2009]), let alone their distribution.

source: http://www.primecoiner.com/prime-numbers-an-odyssey/

Prime Numbers