## Introduction to my Twin Primes Paper

There is a question in Number Theory that has been asked, and not yet answered, for over two millennia. It is quite likely that the great mathematician Euclid (c. 300 B.C.) puzzled over it, although we have no evidence that he did.

The question is quite simply put:“Is the list of twin primes endless?” The Twin Primes Conjecture (abbreviated by TPC) is that the answer is “Yes”. So what is this list … what are these things called ‘twin primes?”

In the endless list of prime numbers, viz. 2, 3, 5, 7, 11, 13, 17,

one can see a sub-list of consecutive pairs of odd numbers, both primes, viz. (3,5), (5,7), (11,13), … Such pairs are called *twin primes*.

Euclid himself proved that the list of prime numbers is endless (we say ‘infinite’). You can read that proof in his famous book *Elements of Geometry*. You can still probably get a copy of the book from Amazon.com!

But so far, to my knowledge, no one has managed to prove that the second list, of the twin primes, is endless too. My friend Paul Bruckman posted some three years ago a paper on this website, which claims to have a proof of the conjecture that twin primes go on occurring for ever (the conjecture is known by the acronym TPC). So far, again to my knowledge, he hasn’t found any top-class mathematician who has read the paper, and has agreed that his TPC proof is valid. Nor has he persuaded any Editor of a Mathematics Journal to publish the paper. Poor Paul! He and I keep trying to achieve one or other of these aims.

Now I am putting forward my own claim to have solved the puzzle, using a completely different line of attack from Paul’s. I firmly believe that my approach can provide a solution, and am now posting my first paper on the matter to my website.

I know that I shall become engulfed in the same difficulties that Paul is facing. So be it. I am proud that I have been able to get thus far with my attempts. I am prepared to carry on with my struggle to prove the TPC for another five and more years — indeed,’til death do us part! In Mathematics, as with other worthwhile endeavours, the journey provides the adventure and the fascinating new insights; the arrival point is a different animal.

Many mathematicians, of greater mental powers than mine, have kept their efforts sizzling, unsuccessfully, on the TPC for much longer than ten years.

If you wish to keep up with my epic struggle to dispose of the TPC, and to find out how Paul is doing too, keep visiting this website from time to time, and following the link below . Maybe you will read through his papers, too. Now *that*’s a real challenge for you … !

Best of luck … John Turner, January, 2012.

(0,1)-PATTERNS GENERATED BY A DOUBLE-CYCLING PROCESS; WITH A PROOF OF THE TWIN PRIMES CONJECTURE – FIFTH DRAFT – Last updated 10th January 2012