Andrica's conjecture states that, for the th Prime Number, the Inequality

holds, where the discrete function is plotted above. The largest value among the first 1000 Primes is for , giving . Since the Andrica function falls asymptotically as increases so a Prime Gap of increasing size is needed at large , it seems likely the Conjecture is true. However, it 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, ... (Sloane's A001223).

**References**

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.

Rivera, C. ``Problems & Puzzles (Conjectures): Andrica's Conjecture.'' http://www.sci.net.mx/~crivera/conjectures/conj_008.htm.

Sloane, N. J. A. Sequence
A001223/M0296
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-25