Andrica's Conjecture
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 aPrime 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 ofwhich 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. SequenceA001223/M0296in ``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.
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- Ramanujan's Hypothesis
- Ramanujan's Identity
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- Random Distribution
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- Random Graph
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