Gilbreath Conjecture

Within the card magic community the Gilbreath Principle is a well-known but much misunderstood mathematical principle. Few magicians know much about its creator, Norman Gilbreath, and in particular they are unaware of his other mathematical work. Following a recent email conversation with him about the principle (always go to the source!) he kindly sent me an offprint of a recent paper on the Gilbreath Conjecture. The Gilbreath Conjecture is a conjecture about primes and is fairly easy to state. Consider the sequence of primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … Now work out the (absolute) difference between neighbouring terms 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, … Work out the absolute difference between terms for this sequence and keep doing this: 1, 0, 2, 2, 2, 2, 2, 2, 4, 4, … 1, 2, 0, 0, 0, 0, 0, 2, 0, … 1, 2, 0, 0, 0, 0, 2, 2, … 1, 2, 0, 0, 0, 2, 0, … 1, 2, 0, 0, 2, 2, … 1, 2, 0, 2, 0, … 1, 2, 2, 2, … 1, 0, 0, … 1, 0, … 1, … The conjecture is that the first term on a line, after the first line, is always a 1. Gilbreath’s paper, Processing process: The Gilbreath conjecture, has recently been published in the Journal of Number Theory. (You can find it at http://dx.doi.org/10.1016/j.jnt.2011.06.008 but unless you have a subscription to the journal you will have to pay for it.) The introduction states the following There is one very important aspect of history that is often left out – the process. This is even true of the history of mathematics. I will give an example. A number of years ago...

Vladimir Zakalyukin Jan03

Vladimir Zakalyukin

At the weekend I heard that Vladimir Zakalyukin had died suddenly in Moscow on 30 December. He was well-known amongst the Singularity Theory community and was a student of Arnold. An expert on symplectic geometry, he was one of the first, if not the first, to classify Lagrangian and Legendrian singularities in depth. He also published work in various areas such as motion planning and symmetry sets. He was research active to the end with a PhD student just finishing a few months ago and was to be an organizer for the Bill Bruce and Terry Wall conference in June. I first met him when he was visiting Warwick University in the early nineties just after the fall of communism. I think he was the first Russian I ever met. More recently I’d often see him as he had an unusual academic position in that he was able to spend spend six months of the year in Liverpool and six in Moscow. Aside from his mathematics I will remember him for his sense of humour (always difficult to communicate in a second language). At one conference dinner he related to us an hilarious story of how he went hunting and ended up being the hunted. He’ll be missed by many. [The above photo was taken by Vera Timofeeva. I attempted to contact her for permission to use it but was unable to find any contact details. If anyone knows them, can they let me know? Thanks. UPDATE: Thanks to Joel Haddley for giving me the details. FURTHER UPDATE: Vera has informed me that Ugo Boscain took the picture. My thanks to him for allowing me to use...

New Year

At the start of each year I always like to review the year just gone and maybe make one or two New Year’s resolutions. Last year’s resolution was to finish things rather than keep starting new things and not finishing them. That resolution didn’t work out – well it nearly did. Some long-standing projects were finally completed and a lot of projects moved into the nearly done phase. I finished my new talk – on autotune. If I had 13 months in the year I could have finished the other stuff too, for example, two research papers need to be completed and my inbox still has too many unanswered emails. It really was a year of ups and downs. (Isn’t it always?) The year started with me accepting an award for my teaching and only days later my application for a pay rise was rejected. On the other hand, my book continued to sell very well and pushed past the 5000 copies sold mark; I appeared in the papers and on TV and radio (even if it was on a rather minor matter, it was still good fun); I produced a DVD (first of many?); I co-edited two proceedings (one of which has appeared). I could list more but, hey, I’m holiday! What about 2012? Well, this year’s resolution is the same as last year – I’m going to finish the leftover stuff. Watch this...