generalized Andrica conjecture

The Andrica function $A_{n}\equiv\sqrt{p_{n+1}}-\sqrt{p_{n}}$ , where $p_{n}$ is the n ${}^{\text{th}}$ prime number can be plotted with mathematical software and for large $n$ it seems that $1\gg A_{n}$ , however the Andrica conjecture $1>A_{n}$ has not been yet proven and remains an open problem.

Similarly one can consider the generalized Andrica function $A_{G}(x,n)\equiv p_{n+1}^{x}-p_{n}^{x}$ and plot it for $x\in\mathbb{R}$ .

It is clear that $A_{G}(0,n)=0$ .

For $x<0$ , $A_{G}(x,n)$ is negative, and if $x\rightarrow-\infty$ then $A_{G}(x,n)\rightarrow-\infty$ .

For $x>0$ , $A_{G}(x,n)$ is positive, and if $x\rightarrow+\infty$ then $A_{G}(x,n)\rightarrow+\infty$ .

Therefore if one considers the generalized Andrica equation $A_{G}(x,n)=1$ and solves for $x$ then solutions for each $n$ will occur for $x>0$ . What is more it is easily provable that the biggest solution of generalized Andrica equation $x_{\max}=1$ occurs for $n=1$ , and for $n>1$ it is always the case that each solution of generalized Andrica equation $x_{n}<1$ because the minimal difference between two consequtive primes is at best 2 for twin primes. However the value of the smallest solution of generalized Andrica equation $x_{\min}$ at the present time remains unknown and its existence is unproven.

The existence of minimal solution $x_{\min}$ of the generalized Andrica equation is still unproven. However according to the generalized Andrica conjecture proposed by Florentin Smarandache the value of $x_{\min}$ , also known as the Smarandache constant, is $x_{\min}\approx 0.5671481302\ldots$ and occurs for $n=30$ . If stated as an inequality the generalized Andrica conjecture states:

$p_{n+1}^{x}-p_{n}^{x}<1$ for $x<0.567148\ldots$

Numerical plots for the first $2\times 10^{11}$ primes show that the solutions $x_{n}$ of $A_{G}(x,n)=1$ tend to be confined in the interval $(0.9,1)$ and according to generalized Andrica conjecture one hopes that this behavior remains true as $n\rightarrow\infty$ .

The following plots of $A_{G}(x,n)$ were created with Wolfram’s Mathematica 5.2, the function plot range was cut off at $A_{G}(x,n)=1$ , so the edge of the plateau is visualizing the exact solutions $x_{n}$ of the equation $A_{G}(x,n)=1$ .

Plots for the first 200 primes. This plot most clearly visualizes the putative minimal solution $x_{\min}$ known also as the Smarandache constant, which seems to occur for $n=30$ .

Plots for the first 1000 primes.

Plots for the first $2\times 10^{3}$ primes.

Plots for the first $2\times 10^{4}$ primes.

Plots for the first $2\times 10^{5}$ primes.

Plots for the first $2\times 10^{6}$ primes.

Plots for the first $2\times 10^{9}$ primes.

Plots for the first $2\times 10^{11}$ primes.

Title	generalized Andrica conjecture
Canonical name	GeneralizedAndricaConjecture
Date of creation	2013-03-22 17:17:34
Last modified on	2013-03-22 17:17:34
Owner	dankomed (17058)
Last modified by	dankomed (17058)
Numerical id	32
Author	dankomed (17058)
Entry type	Conjecture
Classification	msc 11A41
Related topic	FlorentinSmarandache
Related topic	SmarandacheFunction