Gilbreath's conjecture

Gilbreath's conjecture is a number theory problem about prime numbers. Write down all the prime numbers, like so:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

and then write down the absolute difference of subsequent values in the above sequence, and then do the same with the resulting sequence. What you get looks like:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, ...

1, 0, 2, 2, 2, 2, 2, 2, 4, ...

1, 2, 0, 0, 0, 0, 0, 2, ...

1, 2, 0, 0, 0, 0, 2, ...

1, 2, 0, 0, 0, 2, ...

1, 2, 0, 0, 2, ...

Equivalently, let a_n be a value of the original sequence, and b_n be a value of the new sequence; then

b_n = | a_n - a_{n + 1} | .

The Gilbreath conjecture says that the first value of this sequence always equals 1, except for the original sequence of primes. It has been verified for primes up to 10¹³. It is attributed to N. L. Gilbreath, in 1958.