Gilbreath's conjecture

Gilbreath's conjecture is a conjecture in number theory about the effect of difference operators on the sequence of prime numbers. It is named after Norman L. Gilbreath who came up with it in 1958. Long before that François Proth had actually discovered and published this effect in 1878. Proth claimed to have proved it but the proof was not correct. [Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=GilbreathsConjecture The Prime Glossary: Gilbreath's conjecture] at The Prime Pages.]

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 (3-2=1; 5-3=2; 7-5=2; 11-7=4; etc.) 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

:$b\_n\; =\; |a\_n\; -\; a\_\{n+1\}|$.

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 $10^\{13\}$. [A. M. Odlyzko, " [http://www.dtc.umn.edu/~odlyzko/doc/arch/gilbreath.conj.ps Iterated absolute values of differences of consecutive primes] ", Mathematics of Computation, 61 (1993) pp. 373–380. ]

Notes