Often when I am bored in class, or on the bus, I'll take out a piece of scratch paper and "doodle with numbers." It makes me think, even though the things I come up with are practically useless.

For example, today I was messing with lists of prime numbers (with the number 1 included)

Like:
1,2,3,5,7

Now I make a new list where each term is the absolute value of the difference of two adjacent numbers from the previous list. For the list above it would be:
1, 1, 2, 2
and recursively doing the same operation on each list until only one term is left:
0,1,0
1,1
0
So it forms a kind of tree structure.

I noticed if the starting list has n elements, then if nis odd, the final number is 0. If n is even, the final number is 1. I am convinced this is true for all lists of prime numbers (with the number 1 included), even though I have not proved it.

However I don't think this property is unique for lists of prime numbers. I think prime numbers must be a subset of a more general pattern. How can I figure this out?

Sorry if this is the wrong section, but it's not homework so I didn't know where else to put it.

Does anyone else like to play with numbers like this?

edit: I realized a couple minutes after posting that starting with a 1 is is pointless. Instead just take a list of prime numbers and it will reduce to 1, always.

this is an interesting observation and i'm pretty sure that it's not true in general! you just need to check this for larger primes, (maybe not even that large!) probably by getting some help from

computer. i hope someone here will do that, since i'm interested to know the first prime which contradicts your prediction! also let's make your conjecture "longer living" by dropping the odd

and even part and just say that at the end we'll get either 0 or 1. also, to make it clear, i think by starting list, you meant: where is the (n-1)-th prime number.

Yes, I tried that to practice my Java skills

It is true up to somewhere around 5000, which is where I stopped because the calculations get too big for my processor to handle. I'm sure my programming algorithm is inefficient and could be written to be much faster though.

Edit: I revised my code to make it a bit faster, then let it run in for a while. It's still slow, but now I've confirmed my conjecture is true for any set of primes 2, 3 ,5, ... up to the 10000th prime (which is 104,729). Here is the code for anyone interested:
http://codeviewer.org/view/code:7c6

This requires a primes.txt file that lists prime numbers with a space between them.

Still wondering how to prove it though...hmmm...

Apparently this has already been discovered by someone else and studied extensively.

Gilbreath's Conjecture -- from Wolfram MathWorld

It is shown to be true for all primes up to and including pi(10^13).

Well, I'm at least proud of myself for coming up with this independently.