Guess by Andrica

The first 100 values ​​for ${\displaystyle {\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}}$

The conjecture of Andrica , named after Dorin Andrica is a presumption in the prime gaps .

Let the -th prime number . Then Andrica's conjecture says that the following inequality holds for all natural ones : ${\displaystyle p_{n}}$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle {\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}<1.}$

Using the -th prime number gap , it can also be formulated as follows: ${\displaystyle n}$ ${\displaystyle g(n):=p_{n+1}-p_{n}}$

${\displaystyle g(n)<{\sqrt {p_{n+1}}}+{\sqrt {p_{n}}}.}$

values

The first 500 values ​​for . ${\displaystyle A_{n}={\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}}$

Be it . ${\displaystyle A_{n}:={\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}}$

Empirically, these values ​​decrease asymptotically for increasing , so it is very likely that the assumption is correct. For everyone with , H. J. Smith's suggestion was confirmed, which was the greatest value found . ${\displaystyle n}$ ${\displaystyle A_{n}}$ ${\displaystyle n<26\cdot 10^{10}}$ ${\displaystyle A_{4}\approx 0{,}670873479}$

Some values, of which it is assumed that they are no longer exceeded for larger ones, can be found in the following table: ${\displaystyle n}$

${\displaystyle n}$ Follow A084976 in OEIS	${\displaystyle p_{n}}$ Follow A084974 in OEIS	${\displaystyle A_{n}}$ Follow A084977 in OEIS
4th	7th	0.670873
30th	113	0.639281
217	1327	0.463722
263	1669	0.292684
367	2477	0.260522
429	2971	0.256245
462	3271	0.244265
590	4297	0.228429
650	4831	0.215476
738	5591	0.213675
...
10655462	191912783	0.008950

Numerical computer calculations confirm the assumption; meanwhile (2005) the prime numbers were tested to. However, formal proof has not yet been provided. ${\displaystyle 10^{16}}$

generalization

More generally, one can use the equation

${\displaystyle p_{n+1}^{x}-p_{n}^{x}=1}$

consider and look for the maximum or minimum that satisfies such an equation. You have the equation ${\displaystyle x}$

• Maximum trivially at , d. H. ${\displaystyle n=1}$

${\displaystyle 3^{x}-2^{x}=1\qquad \Rightarrow \qquad x=1}$

• Minimum for the first 1000 primes (and presumably general) at , d. H. ${\displaystyle n=30}$

${\displaystyle 127^{x}-113^{x}=1\qquad \Rightarrow \qquad x=0{,}56714813\dots =a_{0}}$

This is (also known as the ) Smarandache constant referred to. ${\displaystyle a_{0}}$

This gives rise to the generalized Andricean conjecture

${\displaystyle B_{n}=p_{n+1}^{a}-p_{n}^{a}<1\qquad {\text{für alle }}a<a_{0}.}$

It is also believed that

${\displaystyle C_{n}=p_{n+1}^{1/k}-p_{n}^{1/k}<{\frac {2}{k}}\qquad {\text{wobei }}k\geq 2,k\in \mathbb {N} ,n\in \mathbb {N} .}$

Similar guess

Andrica's conjecture is a tightening of Legendre's conjecture , according to which there is at least one prime number between each and every one. ${\displaystyle n^{2}}$ ${\displaystyle (n+1)^{2}}$

literature

• Florentin Smarandache : Six Conjectures which Generalize or Are Related to Andrica's Conjecture. In: Octogon. Volume 7, No. 1, 1999, pp. 173-176. arxiv : 0707.2584v1 ; see: Perez

Web links

• Eric W. Weisstein : Andrica's Conjecture . In: MathWorld (English).
• Andrics's Conjecture and Generalized Andrica conjecture on PlanetMath
• ML Perez: Five Smarandache Conjectures on Primes

Individual evidence

1. ^ Titu Andreescu: Number Theory. Springer Science & Business Media, 2009, ISBN 978-0-8176-4645-5 , p. PT26 ( limited preview in Google book search).
2. ↑ Prime Numbers: The Most Mysterious Figures in Math . John Wiley & Sons, 2005, p. 13.
3. ↑ sequence A038458 in OEIS
4. ↑ It is not to be confused with the sixteen Smarandache constants n , which are related to the Smarandache function .