## Five Papers by Paul S. Bruckman

Three and a half years ago, I received an email from Paul which left me amazed, startled and excited for the whole of the day on which I read it.

The short email went through the usual niceties … he asked me how I was, what maths. I was doing, and why I hadn’t written for a while. Then he casually stated that he had had a few ideas recently on how to solve the following problems. I don’t recall the ordering of his list, but I remember they were labelled (a), (b) and (c), with the names Riemann Hypothesis, Goldbach Conjecture, and Twin Primes Conjecture attached to them.

Had you received that email, would not you too have gasped in surprise and disbelief, your hackles rising in excitement? Surely every mathematician, and indeed many others who have had the least brush with mathematics, will have heard those names of open problems. They are all famous, with venerable histories, and thousands of mathematicians, professionals and amateurs, have struggled to solve them.

From that day, Paul and I have maintained a regular email correspondence. He has kept me updated on his progress with attempts to dispose of the three problems. And on the way he has made strong attempts to solve two other well-known mathematical problems, one of which is the so-called Collatz Conjecture (also known as the 3x+1 problem).

My computer folders labelled PaulPapers, PaulWeb or Paul.doc, or some such title, have multiplied over the years, and they are filled with Paul’s many attempts to provide solutions to the ‘great problems’. It has been something of a roller-coaster ride for me, trying to understand each paper as it has arrived, looking at his methods, looking for errors (typos, logical errors etc.), and so on. And those folders kept growing in size, when revisions had to be made, and then perhaps revisions of the revisions; or else logical errors were evaded by pursuing new directions altogether, leading to new papers, and then their revisions and more revisions. Big fleas and little fleas!.

I myself do not have the mathematical power, nor the competences, to finally judge whether any particular paper of Paul’s is ‘right’ … whether he has truly established full proof of the ‘great problem’ concerned. I know how deep the subtleties of the problems run (if that were not so, they would have been solved long ago). However, I have greatly enjoyed my roller-coaster ride, sharing Paul’s adventures and excitements in these rarified pieces of mathematics, being close to the action, as it were. I have admired Paul’s persistence and resilience in tackling the problems, and I have maintained faith in his abilities to finally solve them. Perhaps I should say I have always kept alive the hope that he will solve at least one of them, because a solution of any one will give him a measure of lasting fame in the mathematics world. I do so wish that for him.

During the past three years he has sent at least three of his papers to leading Mathematical Journals. To my knowledge, two of these are still languishing in Journal editors’ files. One was returned as ‘not being a suitable topic for the Journal concerned’. This was surely ludicrous, since it was his purported proof of the Goldbach conjecture! Perhaps the Editor had placed it, unread, in the out-tray labelled ‘Cranks’.

Those of you who have visited this site in the last five months (April to August, 2010) will know that Paul and I finally decided to put four of his papers on my website, and invite people to read them and pass comments and judgments upon them. I did this, opening up a Page entitled ‘Four Great Papers by Paul S. Bruckman’.

I wish to thank those readers who spent time and energy in reading the papers, and drawing Paul’s attention to errors, and corresponding with him over them, and for the kindnesses they expressed towards him in dealing with the issues.

However, a beneficial result of all this website activity is that two or three mathematicians have collaborated with Paul, as he worked out ways and means to correct the errors that were pointed out to him. I have already mentioned his persistence and resilience, which I greatly admire. He will not believe that he should not continue with his attempts to solve ‘the great problems’. After all, given his great, proven abilities as a ‘problem solver’, why should he? I shall continue to keep on this website any paper that he wishes to place there. In future, I shall not re-write the Introduction whenever he withdraws a paper and inserts a revised paper, but simply note, with a date, the changes that have been made.

I hope that some mathematician readers out there will join me in wishing Paul success in his continued efforts. Join me in following Paul’s roller-coaster ride with all its attendant excitements. If we all shout “Tally Ho” loudly enough (to switch metaphors) we shall one day see him catch a fox!

REVISIONS (30th August, 2010)

I have changed the website Menu (Page) title to ‘Five Papers by Paul S. Bruckman’.

The five papers now listed are:

A_NEW_PROOF_OF_THE_COLLATZ_CONJECTURE_(08-28-2010)

ON_THE_OCCURRENCE_OF_PRIMES_2nd_revision_(06-11-2010)

A_NEW_PROOF_OF_THE_STRONG_GOLDBACH_CONJECTURE_(09-07-2010)

A_PROOF_OF_THE_RIEMANN_HYPOTHESIS_revised_(09-05-2010)

A PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS (02-22-2011)