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| |footer=Graphical proof for Andrica's conjecture for the first (a)100, (b)200 and (c)500 prime numbers. The function <math>A_n</math> is always less than 1.
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| |image1=Andrica's Conjecture.svg
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| |caption1=(a) The function <math>A_n</math> for the first 100 primes.
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| |image2=Andrica's Conjecture2.svg
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| |caption2=(b) The function <math>A_n</math> for the first 200 primes.
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| |image3=Andrica's Conjecture3.svg
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| |caption3=(c) The function <math>A_n</math> for the first 500 primes.
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| }}
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| '''Andrica's conjecture''' (named after [[Dorin Andrica]]) is a [[conjecture]] regarding the [[prime gap|gaps]] between [[prime number]]s.<ref>{{cite journal | first=D. | last=Andrica | authorlink= | title=Note on a conjecture in prime number theory | journal=Studia Univ. Babes-Bolyai Math. | volume=31 | year=1986 | number=4 | pages=44–48 | zbl=0623.10030 | issn=0252-1938 }}</ref>
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| The conjecture states that the inequality
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| :<math>\sqrt{p_{n+1}} - \sqrt{p_n} < 1 </math>
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| holds for all <math>n</math>, where <math>p_n</math> is the ''n''<sup>th</sup> prime number. If <math>g_n = p_{n+1} - p_n</math> denotes the n<sup>th</sup> [[prime gap]], then Andrica's conjecture can also be rewritten as
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| :<math>g_n < 2\sqrt{p_n} + 1.</math>
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| == Empirical evidence ==
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| Imran Ghory has used data on the largest prime gaps to confirm the conjecture for <math>n</math> up to 1.3002 x 10<sup>16</sup>.<ref>''Prime Numbers: The Most Mysterious Figures in Math'', John Wiley & Sons, Inc., 2005, p.13.</ref> Using a table of [[Prime_gaps#Numerical_results|maximal gaps]] and the above gap inequality, the confirmation value can be extended to past 1.4 x 10<sup>18</sup>.
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| The discrete function <math>A_n = \sqrt{p_{n+1}}-\sqrt{p_n}</math> is plotted in the figures opposite. The high-water marks for <math>A_n</math> occur for n = 1, 2, and 4, with ''A''<sub>4</sub> ≈ 0.670873..., with no larger value among the first 10<sup>5</sup> 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.
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| == Generalizations ==
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| As a generalization of Andrica's conjecture, the following equation has been considered:
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| :<math> p _ {n+1} ^ x - p_ n ^ x = 1, </math>
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| where <math> p_n </math> is the ''n''<sup>th</sup> prime and ''x'' can be any positive number.
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| The largest possible solution ''x'' is easily seen to occur for <math>n=1</math>, when ''x''<sub>max</sub>=1. The smallest solution ''x'' is conjectured to be ''x''<sub>min</sub> ≈ 0.567148... {{OEIS|id=A038458}} which occurs for ''n'' = 30.
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| This conjecture has also been stated as an [[inequality (mathematics)|inequality]], the generalized Andrica conjecture:
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| :<math> p _ {n+1} ^ x - p_ n ^ x < 1 </math> for <math>x < x_{\min}.</math>
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| == See also ==
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| * [[Cramér's conjecture]]
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| * [[Legendre's conjecture]]
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| * [[Firoozbakht’s conjecture]]
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| == References and notes ==
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| {{reflist}}
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| * {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 }}
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| == External links ==
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| * [http://planetmath.org/encyclopedia/AndricasConjecture.html ''Andrica's Conjecture''] at [[PlanetMath]]
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| * [http://planetmath.org/?op=getobj&from=objects&id=9636 ''Generalized Andrica conjecture''] at [[PlanetMath]]
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| * {{MathWorld|urlname=AndricasConjecture|title=Andrica's Conjecture}}
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| [[Category:Conjectures about prime numbers]]
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Oscar is how he's known as and he completely enjoys this name. What I adore doing is to gather badges but I've been using on new issues recently. South Dakota is exactly where I've always been residing. My working day occupation is a meter reader.
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