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).
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." https://www.gallup.unm.edu/~smarandache/conjprim.txt.
Rivera, C. "Problems & Puzzles: Conjecture 008.-Andrica's Conjecture." https://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.
