Andrica's Conjecture


Andrica's conjecture states that, for p_n the nth prime number, the inequality


holds, where the discrete function A_n is plotted above. The high-water marks for A_n occur for n=1, 2, and 4, with A_4=sqrt(11)-sqrt(7) approx 0.670873, with no larger value among the first 10^5 primes. Since the Andrica function falls asymptotically as n increases, a prime gap of ever increasing size is needed to make the difference large as n becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven.


A_n bears a strong resemblance to the prime difference function, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, ... (OEIS A001223).

A generalization of Andrica's conjecture considers the equation


and solves for x. The smallest such x is x approx 0.567148 (OEIS A038458), known as the Smarandache constant, which occurs for p_n=113 and p_(n+1)=127 (Perez).

See also

Brocard's Conjecture, Cramér Conjecture, Good Prime, Fortunate Prime, Pólya Conjecture, Prime Difference Function, Smarandache Constants, Twin Peaks

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Andrica, D. "Note on a Conjecture in Prime Number Theory." Studia Univ. Babes-Bolyai Math. 31, 44-48, 1986.Golomb, S. W. "Problem E2506: Limits of Differences of Square Roots." Amer. Math. Monthly 83, 60-61, 1976.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 21, 1994.Perez, M. L. (Ed.). "Five Smarandache Conjectures on Primes.", C. "Problems & Puzzles: Conjecture 008.-Andrica's Conjecture.", N. J. A. Sequences A001223/M0296 and A038458 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. Prime Numbers: The Most Mysterious Figures in Math. New York: Wiley, p. 13, 2005.

Cite this as:

Weisstein, Eric W. "Andrica's Conjecture." From MathWorld--A Wolfram Web Resource.

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