Gilbreath conjecture

2020 Mathematics Subject Classification: Primary: 11A41 [MSN][ZBL]

A conjecture on the distribution of prime numbers.

For any sequence $(x_n)$, define the absolute difference sequence $\delta^1_n = |x_{n+1} - x_n|$, and the iterated differences $\delta^{k+1} = \delta^1 \delta^k$. In 1958 N. L. Gilbreath conjectured that when applied to the sequence of prime numbers, the first term in each iterated sequence $\delta^k$ is always $1$. Odlyzko has verified the conjecture for the primes $\le 10^{13}$.

References

[1]	Andrew M. Odlyzko, "Iterated absolute values of differences of consecutive primes", Math. Comput. 61, no.203 (1993) pp.373-380 DOI 10.2307/2152962 Zbl 0781.11037
[2]	Richard K. Guy, "Unsolved problems in number theory" (3rd ed.) Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001
[3]	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