Gilbreath's Conjecture

Let the Difference of successive Primes be defined by $d_n\equiv p_{n+1}-p_n$, and $d_n^k$ by

$$d_n^k\equiv\cases{ d_n & for $k=1$\cr \vert d_{n+1}^{k-1}-d_n^{k-1}\vert & for $k>1$.\cr}$$

N. L. Gilbreath claimed that $d_1^k=1$ for all $k$ (Guy 1994). It has been verified for $k<63419$ and all Primes up to $\pi(10^{13})$, where $\pi$ is the Prime Counting Function.

References

Guy, R. K. ``Gilbreath's Conjecture.'' §A10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 25-26, 1994.