A Proof of the Twin Prime Conjecture

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.