 TOPICS  # Andrica's Conjecture Andrica's conjecture states that, for the th prime number, the inequality holds, where the discrete function is plotted above. The high-water marks for occur for , 2, and 4, with , with no larger value among the first primes. Since the Andrica function falls asymptotically as increases, a prime gap of ever increasing size is needed to make the difference large as becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven.  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 . The smallest such is (OEIS A038458), known as the Smarandache constant, which occurs for and (Perez).

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

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## References

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." http://www.gallup.unm.edu/~smarandache/conjprim.txt.Rivera, C. "Problems & Puzzles: Conjecture 008.-Andrica's Conjecture." http://www.primepuzzles.net/conjectures/conj_008.htm.Sloane, 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. https://mathworld.wolfram.com/AndricasConjecture.html