# 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 GeneralizedAndricaConjecture 2013-03-22 17:17:34 2013-03-22 17:17:34 dankomed (17058) dankomed (17058) 32 dankomed (17058) Conjecture msc 11A41 FlorentinSmarandache SmarandacheFunction