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Gilbreath's Conjecture

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Gilbreath's conjecture states that d_1^k=1 for all k>=1 in the following array of differences (Guy 1994). Let the difference of successive primes be defined by d_n=p_(n+1)-p_n, and d_n^k by

d_n^k={d_n for k=1; |d_(n+1)^(k-1)-d_n^(k-1)| for k>1.
(1)

N. L. Gilbreath claimed that d_1^k=1 for all k (Guy 1994). Kilgrove and Ralston (1959) verified the claim for k<63419. Odlyzko (1993) verified it for the initial segment containing all primes below 10^(13). Colonna (2026) reports extending this computational verification through 1.5×10^(15), completed on March 18, 2026.

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

2,3,5,7,11,13,17,19,23,29,... 1,2,2,4,2,4,2,4,6,... 1,0,2,2,2,2,2,2,... 1,2,0,0,0,0,0,... 1,2,0,0,0,0,... 1,2,0,0,0,... 1,2,0,0,... 1,2,0,... 1,2,... 1,...
(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, 24, 23, 22, 25, ... (OEIS A000232).

See also

Prime Difference Function

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References

Caldwell, C. K. "Gilbreath's Conjecture." https://t5k.org/glossary/page.php?sort=GilbreathsConjecture.Colonna, J.-F. "Proth-Gilbreath Conjecture Verification up to 1.5×10^(15)." Mar. 18, 2026. https://www.lactamme.polytechnique.fr/Mosaic/descripteurs/Proth_Gilbreath_Conjecture.01.Ang.html.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. Math. 4, 236-240, 1878.Sloane, N. J. A. Sequences A000232/M2718 and A036262 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Gilbreath's Conjecture

Cite this as:

Weisstein, Eric W. "Gilbreath's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GilbreathsConjecture.html

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