The traditional definition of the twin prime conjecture is that there is an infinite number of twin primes. The traditional definition of a twin prime is a pair of odd primes separated by one even number, e.g., 29 and 31.
The twin prime conjecture is a long standing problem. In
many introductory textbooks on number theory, the author has a section (usually
in the first chapter) on open problems. The Twin Prime Conjecture is usually
listed1.. Britannica has a comment on it. Go to "britannica.com" and
search on "twin prime conjecture (number theory)". From the article:
"The first statement of the twin prime conjecture was given in 1846 by French
mathematician Alphonse de Polignac, ..." Some mathematicians have
suggested that Euclid (circa 300 BC) hinted at the twin prime conjecture when
he did his work on the infinitude of the primes.
We give a distinctly new approach to the twin prime conjecture.
We don't use the methods of analytic number theory. Instead, we use
Eratosthenes' sieve2 . We are not interested in the primes that are uncovered.
Instead, we find in the sparse sequences of natural numbers that remain after
each implementation of the sieve, useful structures that we call Eratosthenes'
Patterns. These structures reveal a number of twin primes, and with each
implementation of the sieve, these structures reveal increasing numbers of twin
primes.
The essence of our proof is to show that the number of twin
primes between pn and
approaches infinity
as n approaches infinity.
Moreover, we have found a second constellation of primes,
two primes that differ by four. In each of the Eratosthenes Patterns these
primes are equal in number to the twin primes. We suggest that these are as
significant as the twin primes. They are also infinite in number.
In order, we describe Eratosthenes' Sieve, Eratosthenes'
Patterns, and the twins. Then we give the proof.
Author (s) Details
A. W. Draut
Embry Riddle Aeronautical University, Prescott, AZ, USA.
Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v5/2386
