generalized Andrica conjecture

The Andrica function Anpn+1-pn, where pn is the nth prime numberMathworldPlanetmath can be plotted with mathematical software and for large n it seems that 1An, however the Andrica conjectureMathworldPlanetmath 1>An has not been yet proven and remains an open problem.

Similarly one can consider the generalized Andrica function AG(x,n)pn+1x-pnx and plot it for x.

It is clear that AG(0,n)=0.

For x<0, AG(x,n) is negative, and if x- then AG(x,n)-.

For x>0, AG(x,n) is positive, and if x+ then AG(x,n)+.

Therefore if one considers the generalized Andrica equation AG(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 xmax=1 occurs for n=1, and for n>1 it is always the case that each solution of generalized Andrica equation xn<1 because the minimal difference between two consequtive primes is at best 2 for twin primesMathworldPlanetmath. However the value of the smallest solution of generalized Andrica equation xmin at the present time remains unknown and its existence is unproven.

The existence of minimal solution xmin of the generalized Andrica equation is still unproven. However according to the generalized Andrica conjecture proposed by Florentin Smarandache the value of xmin, also known as the Smarandache constant, is xmin0.5671481302 and occurs for n=30. If stated as an inequality the generalized Andrica conjecture states:

pn+1x-pnx<1 for x<0.567148

Numerical plots for the first 2×1011 primes show that the solutions xn of AG(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.

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

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

Plots for the first 1000 primes.

Plots for the first 2×103 primes.

Plots for the first 2×104 primes.

Plots for the first 2×105 primes.

Plots for the first 2×106 primes.

Plots for the first 2×109 primes.

Plots for the first 2×1011 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