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# 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.

## Mathematics Subject Classification

11A41*no label found*

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## Corrections

do you want x>1? by Mathprof ✓

despam by mps ✓

reword a bit by Mathprof ✓

## Comments

## How to attach images?

I hope someone will explain me why if I want to attach .emf or .png image the file is not attached to the article, but the attached files area remains empty. Please help with some instructions how to upload plots, and how to insert them in the article. Thanks!

## Re: How to attach images?

It needs to be an .EPS, or better yet, TeX vector instructions. See

http://planetmath.org/?op=getobj&from=collab&id=35

the last section, "How do I include diagrams in my entry?"

http://planetmath.org/?op=getobj&from=collab&id=53

and

http://planetmath.org/?op=getobj&from=collab&id=28

Section 4.8

for further guidance.

## Too many places to look for images instructions?

The parent entry by PrimeFan points the requestor at *3* different places for information about using graphics and images on PlanetMath. This seems like two too many.

It would make sense to me to merge the information from

http://planetmath.org/?op=getobj&from=collab&id=35

and

http://planetmath.org/?op=getobj&from=collab&id=28

into the collaboration on Graphics and PlanetMath and then simply refer to it.

I don't have the time right now to do this, but if it seems like a good idea, I'll put it on my list.

Roger

## And Too many BUGS?

Hello thanks, but

-- The File box manager with "Browse" and "Upload" does not work for unknown reason on my PC.

-- I have however uploaded the .eps files at Geocities, then grab URL, and this time the files got attached to my PlanetMath entry :-)

-- I have exported from Mathematica .eps plots yet \includegraphics{nameoffigure} of the attached .eps files does not work and error appears. Now my plots are attached to the entry, but I cannot display them.

Can someone explain what is going on? Why in Wikipedia the upload and display of images is plain easy, and here the system is full of bugs. The whole system is not user friendly and it is not because of the latex style, because in Wikipedia I also use latex, but no such problems :-)) Any help is appreciated.

I one to learn at least one sure method for displaying images in PlanetMath.

## Re: And Too many BUGS?

Perhaps you should tell us what, exactly, you tried instead of simply complaining about the quality of the software. I think you'd find that there are people willing to help.

You might also try looking at other entries that have eps files displayed. One of my recent entries is ProofOfParallelogramTheorems.

You should look specifically at the preamble as well as the entry text.

## Re: And Too many BUGS?

I didn't notice that the package

\usepackage{graphicx}

has been turned off by % sign.

Why by default this package is turned off???

Seems to me very odd decision for default settings.

## Re: And Too many BUGS?

> Why by default this package is turned off???

Because not every entry has graphics. It is wasteful to

load a specific graphics package every time which is only

used by a few entries.

## Re: And Too many BUGS?

> -- I have however uploaded the .eps files at Geocities, then grab URL, and this time the files got attached to my PlanetMath entry :-)

I too have done this and it works for me most of the time.

> -- The File box manager with "Browse" and "Upload" does not work for unknown reason on my PC.

What happens in my experience is that if I try to upload the file directly from my computer, the file does get uploaded, but the whole path becomes the filename, for example, C:WindowsUsersJohnPicturesMandelbrotSet.eps. I haven't tried uploading from my Mac, though.