# Is there any progress toward solving Gilbreath's conjecture?

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.

• What is the conjecture? – José Figueroa-O'Farrill Aug 5 '10 at 18:30
• 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 $10^{11}$. – Joseph O'Rourke Aug 5 '10 at 18:31
• 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 – Joseph O'Rourke Aug 5 '10 at 19:01
• 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. – Srilakshmi Aug 19 '12 at 6:15
• 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 – Gerry Myerson Aug 16 '17 at 13:18