# 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 (around 1958) I developed a number theory conjecture concerning the primes. This is known in number theory as the Gilbreath conjecture. It is easy to state but even though the great number theorist Erdos believed it was true, he also believed it would take about 200 years to prove.

Who am I to doubt Erdos? However, it seems no one has seriously considered why and how I
came up with this conjecture. So, I will now describe the process, and some observations suggested by this process – to hopefully show why processing process is important.

If you are interested in the conjecture, then see Wikipedia, Facebook, Wolfram demonstrations, and a request for help by a poster on the xkcd forum.