## 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...

Jan03