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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.

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    $\begingroup$ What is the conjecture? $\endgroup$
    – José Figueroa-O'Farrill
    Aug 5, 2010 at 18:30
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    $\begingroup$ It seems you didn't state the conjecture: "the first term in each series of differences appears to be 1" according to Wikipedia. That article says, "As of 2009, no valid proof of the conjecture has been published." It has been verified up to $10^{11}$. $\endgroup$
    – Joseph O'Rourke
    Aug 5, 2010 at 18:31
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    $\begingroup$ Incidentally, when you are quoting verbatim from a Wikipedia article, it would be appropriate to make that clear. en.wikipedia.org/wiki/Gilbreath%27s_conjecture $\endgroup$
    – Joseph O'Rourke
    Aug 5, 2010 at 19:01
  • $\begingroup$ When i came to know abt. this conjecture, i thought first row p1, p2, p3, p4,....and second row p2-p1, p3-p2, p4-p3,... and third row p3-2p2+p1...and i ended up with pascal's triangle. i.e. To Prove (n-1)C0 p_n - (n-1)C1 p_(n-1) +...+(-1)^(n-1) (n-1)C1 p1 = 1. This might be done by applying a formula for p_n (for eg. paper by willans ). Then i realised that i forgot about the absolute values of the differences. $\endgroup$
    – Srilakshmi
    Aug 19, 2012 at 6:15
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    $\begingroup$ There's a paper by Gilbreath on the conjecture, but I haven't seen it: Norman Gilbreath, "Processing process: the Gilbreath conjecture", J. Number Theory 131 (2011) pp.2436-2441 DOI 10.1016/j.jnt.2011.06.008 Zbl 1254.11006 $\endgroup$
    – Gerry Myerson
    Aug 16, 2017 at 13:18

1 Answer 1

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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.

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    $\begingroup$ That's a great quote! $\endgroup$
    – Andy Putman
    Aug 5, 2010 at 22:10
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    $\begingroup$ The Odlyzko paper is Math. Comp. 61 (1993) 373-380, MR 93k:11119. The conjecture is problem A10 in Guy, Unsolved Problems In Number Theory. In 1878, long before Gilbreath made the conjecture (1958, unpublished), Proth claimed to have proved it - Guy gives the bibliographic details. Odlyzko discusses the suggestion that the result is true for (quoting Guy) any sequence consisting of 2 and odd numbers, which doesn't increase too fast, or have too large gaps. Math Reviews conatins no citations of Odlyzko's paper. $\endgroup$
    – Gerry Myerson
    Aug 5, 2010 at 23:12
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    $\begingroup$ The question of how many increasing sequences of integers have the Gilbreath property is answered in oeis.org/classic/A080839. It doesn't make the primes seem that special. $\endgroup$
    – tdnoe
    Aug 6, 2010 at 5:22
  • $\begingroup$ They're still rare in the sense that A080839 is small compared to A136465, though. $\endgroup$
    – Charles
    Aug 9, 2010 at 3:46

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