The twin primes

Abstract

The primes, including the twin primes and the other prime pairs, are the building-blocks of the integers. Euclid's proof of the infinitude of the primes is generally regarded as elegant. It is a proof by contradiction, or,  reductio ad absurdum, and it relies on an algorithm which always brings in larger and larger primes, an infinite number of them. However, the proof is also subtle and is misinterpreted by some, with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them (see, B. Wong, Possible solutions for the ``twin'' primes conjecture – The infinity of the twin primes, International Mathematical Journal, 3(8), 2003, 873--886). This informative paper presents some important facts on the twin primes which would be of interest to prime number researchers, with some remarks/reasons that point to the infinitude of the twin primes, including a reasoning which is somewhat similar to Euclid's proof of the infinity of the primes; very importantly, two algorithms are developed in Section 5 for sieving out the twin primes from the infinite list of the integers, which would be of interest to cryptographers and even to computer programmers.