Gilbreath's conjecture

Norman Gilbreath's conjecture is a conjecture in number theory about the effect of difference operators on the sequence of prime numbers, attributed to Norman L. Gilbreath, in 1958.

Problem definition

Write down all the prime numbers, thus:

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

$b_{n}=|a_{n}-a_{n+1}|$.

Norman Gilbreath's conjecture states that the first value of this sequence always equals 1, except in the original sequence of primes. It has been verified for primes up to $10^{13}$.