Not really very surprising, as Hallard Croft and others have pointed out. At the foot of this message is the present state of UPINT A10. Have a look at Odlyzko's paper. Note that it should be called Proth's `theorem'! R. On Tue, 1 Jul 2003, Marc LeBrun wrote:
What is the current state of understanding, if any, of Gilbreath's Conjecture?
(This is that every row after the first begins with a 1 in the following
2 3 5 7 11 13 17 19 23 29 31 ... 1 2 2 4 2 4 2 4 6 2 6 ... 1 0 2 2 2 2 2 2 4 4 2 ... 1 2 0 0 0 0 0 2 0 2 0 ... 1 2 0 0 0 0 2 2 2 2 0 ... 1 2 0 0 0 2 0 0 0 2 0 ... 1 2 0 0 2 2 0 0 2 2 2 ... 1 2 0 2 0 2 0 2 0 0 0 ... 1 2 2 2 2 2 2 2 0 0 0 ... 1 0 0 0 0 0 0 2 0 0 2 ... 1 0 0 0 0 0 2 2 0 2 0 ... ...
where the top row is the primes and each lower row the absolute differences of that above it. It has been extensively computationally verified.)
This seems MUCH more remarkable than, say, Goldbach's Conjecture!
\usection{A10}{Gilbreath's conjecture.} Define $d^k_n$ by $d^1_n=d_n$ and $d_n^{k+1}=|d_{n+1}^k-d_n^k|$, that is, the successive absolute differences of the sequence of primes (Figure 2). N.~L.~Gilbreath conjectured (and Hugh Williams notes that Proth long before claimed to have proved)\hGidx{Gilbreath's conjecture} that $d^k_1=1$ for all $k$. This was verified for $k<63419$ by Killgrove \& Ralston. Odlyzko has checked it for primes up to $\pi(10^{13})\approx 3\cdot10^{11}$; he only needed to examine the first 635 differences. \input{fig2} % much as you've got in your message above. \hGidx{Croft, Hallard T.} Hallard Croft and others have suggested that it has nothing to do with primes as such, but will be true for any sequence consisting of 2 and odd numbers, which doesn't increase too fast, or have too large gaps. Odlyzko discusses this. \medskip \small \Aidx{Killg}\Aidx{Ralst} R.~B.~Killgrove \& K.~E.~Ralston, On a conjecture concerning the primes, {\it Math.\ Tables Aids Comput.}, {\bf13}(1959) 121--122; {\it MR} {\bf21} \#4943. \Aidx{Odlyz} Andrew M.~Odlyzko, Iterated absolute values of differences of consecutive primes, {\it Math.\ Comput.}, {\bf61}(1993) 373--380; {\it MR} {\bf93k}:11119. \Aidx{Proth} F.~Proth, Sur la s\'erie des nombres premiers, {\it Nouv.\ Corresp.\ Math.}, {\bf4}(1878) 236--240. \normalsize