Make a triangle of numbers in which the leftmost column is the sequence of prime numbers and each other number is the absolute value of the difference of the two numbers to its left:

2
3 1
5 2 1
7 2 0 1
11 4 2 2 1
13 2 2 0 2 1
17 4 2 0 0 2 1
19 2 2 0 0 0 2 1
23 4 2 0 0 0 0 2 1
29 6 2 0 0 0 0 0 2 1
31 2 4 2 2 2 2 2 2 0 1
37 6 4 0 2 0 2 0 2 0 0 1
41 4 2 2 2 0 0 2 2 0 0 0 1
43 2 2 0 2 0 0 0 2 0 0 0 0 1
47 4 2 0 0 2 2 2 2 0 0 0 0 0 1
53 6 2 0 0 0 2 0 2 0 0 0 0 0 0 1
59 6 0 2 2 2 2 0 0 2 2 2 2 2 2 2 1
61 2 4 4 2 0 2 0 0 0 2 0 2 0 2 0 2 1
67 6 4 0 4 2 2 0 0 0 0 2 2 0 0 2 2 0 1
71 4 2 2 2 2 0 2 2 2 2 2 0 2 2 2 0 2 2 1
73 2 2 0 2 0 2 2 0 2 0 2 0 0 2 0 2 2 0 2 1
79 6 4 2 2 0 0 2 0 0 2 2 0 0 0 2 2 0 2 2 0 1
83 4 2 2 0 2 2 2 0 0 0 2 0 0 0 0 2 0 0 2 0 0 1
89 6 2 0 2 2 0 2 0 0 0 0 2 2 2 2 2 0 0 0 2 2 2 1
97 8 2 0 0 2 0 0 2 2 2 2 2 0 2 0 2 0 0 0 0 2 0 2 1

Notice the pattern of 1’s running down the right side? Gilbreath’s conjecture states that it continues like that forever.

If you want to see proof of another sequence of integers that continues forever, go to http://11011110.livejournal.com/212920.html and you will find the Python source code used to make it happen and the reasoning behind it.