N. L. Gilbreath claimed that for all (Guy 1994). In 1959, the claim was verified for . In 1993, Odlyzko extended the claim to all primes
up to .
Gilbreath's conjecture is equivalent to the statement that, in the triangular array of the primes, iteratively taking the absolute
difference of each pair of terms
(2)
(OEIS A036262), always gives leading term 1
(after the first row).
The number of terms before reaching the first greater than two in the second, third,
etc., rows are given by 3, 8, 14, 14, 25, 23, 22, 25, ... (OEIS A000232).
Caldwell, C. K. "Gilbreath's Conjecture." http://primes.utm.edu/glossary/page.php?sort=GilbreathsConjecture.Debono,
A. N. "Numbers and Computers (11): More on Primes." http://www.eng.um.edu.mt/~andebo/numbers/numcom11.htm.Gardner,
M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers."
Sci. Amer.243, 18-28, Dec. 1980.Guy, R. K. "Gilbreath's
Conjecture." §A10 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 25-26,
1994.Kilgrove, R. B. and Ralston, K. E. "On a Conjecture
Concerning the Primes." Math. Tables Aids Comput.13, 121-122,
1959.Odlyzko, A. M. "Iterated Absolute Values of Differences
of Consecutive Primes." Math. Comput.61, 373-380, 1993.Proth,
F. "Sur la série des nombres premiers." Nouv. Corresp. Math4,
236-240, 1878.Sloane, N. J. A. Sequences A000232/M2718
and A036262 in "The On-Line Encyclopedia
of Integer Sequences."
,
and 
by
for all 
(Guy 1994). In 1959, the claim was verified for 
. In 1993, Odlyzko extended the claim to all primes
up to 
.
