Andrica's conjecture

Andrica's conjecture (named after Dorin Andrica) is a conjecture regarding the gaps between prime numbers. [ D. Andrica, "Note on a conjecture in prime number theory." Studia Univ. Babes-Bolyai Math. 31 (1986), no. 4, 44--48. ]

The conjecture states that the inequality:

:holds for all $n$, where $p\_n$ is the $n$^th prime number.If $g\_n\; =\; p\_\{n+1\}\; -\; p\_n$ denotes the n^th prime gap, then Andrica's conjecture can also be rewritten as:

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for $n$ up to 1.3002 x 10¹⁶. ["Prime Numbers: The Most Mysterious Figures in Math", John Wiley & Sons, Inc., 2005, p.13.]

The discrete function $A\_n\; =\; sqrt\{p\_\{n+1\; -\; sqrt\{p\_n\}$ is plotted in the figures opposite. The high-water marks for $A\_n$ occur for n = 1, 2, and 4, with "A"₄$approx$ 0.670873 ..., with no larger value among the first 10⁵ primes. Since the Andrica function decreases 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.

Generalizations

As a generalization of Andrica's conjecture, the following equation has been considered::$p\; \_\; \{n+1\}\; ^\; x\; -\; p\_\; n\; ^\; x\; =\; 1,$where $p\_n$ is the $n$^th prime and "n" can be any positive integer.

The largest possible solution $x$ is easily seen to occur for $n=1$, when "x"_max=1. The smallest solution $x$ is conjectured to be "x"_min $approx$ 0.567148 ... OEIS|id=A038458, known as the Smarandache constant, which occurs for $n=30$. [M.L.Perez. [http://www.gallup.unm.edu/~smarandache/conjprim.txt Five Smarandache Conjectures on Primes] ]

This conjecture has also been stated as a conjectural inequality, the "generalized Andrica conjecture":: for

See also

* Cramér's conjecture

References and notes

External links

* [http://planetmath.org/encyclopedia/AndricasConjecture.html "Andrica's Conjecture"] at PlanetMath
* [http://planetmath.org/?op=getobj&from=objects&id=9636 "Generalized Andrica conjecture"] at PlanetMath
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