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 numbercan be plotted with mathematical software and for large $n$ it seems that $1\gg A_n$, howeverthe Andrica conjecture $1>A_n$ has not been yet proven andremains 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 $x\in ℝ$.

It is clear that $A_G(0,n)=0$.

For , $A_G(x,n)$ is negative, and if $x\to -\mathrm{\infty }$ then $A_G(x,n)\to -\mathrm{\infty }$.

For $x>0$, $A_G(x,n)$ is positive, and if $x\to +\mathrm{\infty }$ then $A_G(x,n)\to +\mathrm{\infty }$.

Therefore if one considers the generalized Andrica equation$A_G(x,n)=1$ and solves for $x$ then solutions for each $n$ will occurfor $x>0$. 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 because the minimal difference between two consequtive primes is atbest 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 $x_{\min }\approx 0.5671481302\mathrm{\dots }$ and occurs for $n=30$. If stated as an inequality the generalized Andrica conjecture states:

for

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 $n\to \mathrm{\infty }$.

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.