Andrica's conjecture

{{Short description|Conjecture about gaps between prime numbers}}

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|footer=Graphical proof for Andrica's conjecture for the first (a)100, (b)200 and (c)500 prime numbers. It is conjectured that the function $A_n$ is always less than 1.

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|image1=Andrica's Conjecture.svg

|caption1=(a) The function $A_n$ for the first 100 primes.

|image2=Andrica's Conjecture2.svg

|caption2=(b) The function $A_n$ for the first 200 primes.

|image3=Andrica's Conjecture3.svg

|caption3=(c) The function $A_n$ for the first 500 primes.

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Andrica's conjecture (named after Romanian mathematician Dorin Andrica (es)) is a conjecture regarding the gaps between prime numbers.{{cite journal | first=D. | last=Andrica | 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 }}

The conjecture states that the inequality

: $\sqrt{p_{n+1}} - \sqrt{p_n} < 1$

holds for all $n$ , where $p_n$ is the nth prime number. If $g_n = p_{n+1} - p_n$ denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

: $g_n < 2\sqrt{p_n} + 1.$

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for $n$ up to {{val|1.3002|e=16}}.{{cite book | last=Wells | first=David | title=Prime Numbers: The Most Mysterious Figures in Math | publisher=Wiley | publication-place=Hoboken (N.J.) | date=May 18, 2005 | isbn=978-0-471-46234-7 | page=13}} Using a more recent table of maximal gaps, the confirmation value can be extended exhaustively to {{val|2|e=19}} > 2⁶⁴.

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 A₄ ≈ 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

File:Generalized Andrica's conjecture.svg

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 nth prime and x can be any positive number.

The largest possible solution for x is easily seen to occur for n=1, when x_max = 1. The smallest solution for x is conjectured to be x_min ≈ 0.567148... {{OEIS|id=A038458}} which occurs for n = 30.

This conjecture has also been stated as an inequality, the generalized Andrica conjecture:

: $p _ {n+1} ^ x - p_ n ^ x < 1$ for $x < x_{\min}.$

References and notes

{{cite book |last=Guy | first=Richard K. | author-link=Richard K. Guy | title=Unsolved problems in number theory | publisher=Springer-Verlag |edition=3rd | year=2004 |section=Section A8|isbn=978-0-387-20860-2 | zbl=1058.11001 }}

External links

[https://web.archive.org/web/20070711030648/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
https://drive.google.com/file/d/1jWbjCbTn5Tf0twXJJochRt7ONwqs6l3D/view?usp=sharing

Category:Conjectures about prime numbers

Category:Unsolved problems in number theory

Andrica's conjecture

Empirical evidence

Generalizations

See also

References and notes

External links