Is there any progress toward solving Gilbreath's conjecture?

Question

Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, Gilbreath observed a pattern while playing with the ordered sequence of prime numbers

2,3,5,7,11,13,17,19,23,29,31,... Computing the absolute value of the difference between the (n + 1)-th and n-th terms in this sequence yields the sequence

1,2,2,4,2,4,2,4,6,2,... If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ad infinitum for each sequence that is the output of such a calculation, the first five sequences in this list are given by

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

Gilbreath claims that the first term in each series of differences appears to be 1.

I want to know recent research toward proving this conjecture.

Answer 1

There is some theoretical progress towards the conjecture in

Chase, Zachary, A random analogue of Gilbreath’s conjecture, ZBL07808058.

If one models the prime gaps $p_{n+1}-p_n$ (beyond the first gap $p_2-p_1=1$) as an even number between $2$ and $2f(n)$ chosen uniformly and independently at random for some slowly growing function, then in this paper the analogue of Gilbreath's conjecture is established almost surely for sufficiently large $n$ provided that $f(n)$ grows slower than $\frac{1}{100} \frac{\log\log n}{\log\log\log n}$. For comparison, the Cramér random model roughly corresponds to taking $f(n) = \log n$ (and also replacing the uniform distribution by an exponential distribution of the same mean). So this is not yet a fully satisfactory heuristic justification towards the conjecture (on the level of, say, the heuristic justification of the twin prime conjecture based on Cramér type models), but is at least a good first step in that direction.

This paper also dug into the assertion that Proth had claimed a proof of this conjecture, but it appears that this assertion was based on a misreading of the text and has since been retracted by its originator.

EDIT: I guess this would be a good place to mention some of the ideas of the proof. Consider a large block of random gaps and take successive absolute value differences. If one can get these differences down to 2 or less then one is done, and the maximum size of the differences is non-increasing as one moves from one row to the next, so the only way one can get "stuck" is if the maximum difference stays at some level $2d>2$ for a very long time. This turns out to mean that some rows contain very long blocks that consist only of $0$ and $2d$. This is easy to rule out with high probability if $d$ is even, as successive differences modulo 4 are tractable to compute. For odd $d$ one has to work harder. After using a Cauchy-Schwarz argument to make these blocks slightly longer (with some positive probability), and again using the control of differences mod 4, the author then shows that the $0$ value must occur with some reasonably large frequency at every row. This turns out to make it very unlikely for long blocks of $0$ and $2d$ to form (as the row just before the first appearance of such a block would create a long block without any $0$s whatsoever).

Answer 2

I don't think that there has been any progress on this conjecture except for numeric verification (see Odlyzko, A. M. "Iterated Absolute Values of Differences of Consecutive Primes", doi:10.2307/2152962, MR 93k:11119), and if you search for publications with the appropriate key words you won't find any more papers after that. It just seems that it is one of those conjectures that are easy to come up with but that are too distant from the rest of mathematics to have a decent chance to be solved in the near future. In this blog post I found the following amusing line

Paul Erdős speculated that Gilbreath’s conjecture is true but it would be 200 years before anyone could prove it. I find Erdős’s conjecture more interesting than Gilbreath’s conjecture.

Another conjecture that would possibly shed some light on this is that the result isn't actually related to prime numbers but it holds for any sequence of appropriate growth rate, but I don't know what the opinion of the specialists is.