Gilbreath's Conjecture

Let the difference of successive primes be defined by , and  by

(1)

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).

REFERENCES:

Caldwell, C. K. "Gilbreath's Conjecture." https://primes.utm.edu/glossary/page.php?sort=GilbreathsConjecture.

Debono, A. N. "Numbers and Computers (11): More on Primes." https://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. Math 4, 236-240, 1878.

Sloane, N. J. A. Sequences A000232/M2718 and A036262 in "The On-Line Encyclopedia of Integer Sequences."