# 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 $$, ${A}_{G}(x,n)$ 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 occur
for $x>0$. What is more it is easily provable that the biggest solution of generalized Andrica equation ${x}_{\mathrm{max}}=1$
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 at
best 2 for *twin primes ^{}*. However the value of the smallest solution of generalized Andrica equation
${x}_{\mathrm{min}}$ at the present time remains unknown and its existence is unproven.

The existence of minimal solution ${x}_{\mathrm{min}}$ of the generalized Andrica equation is still unproven. However according to the *generalized Andrica conjecture* proposed by Florentin Smarandache the value of ${x}_{\mathrm{min}}$, also known as the *Smarandache constant*, is ${x}_{\mathrm{min}}\approx 0.5671481302\mathrm{\dots}$ 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 ${x}_{\mathrm{min}}$ 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 |