generalized Andrica conjecture

generalized Andrica conjecture

The

Andrica function

A

n

≡

p

n

+

1

-

p

n

, where

p

n

is the n

th

prime number

can be plotted with mathematical software and for large

n

it seems that

1

≫

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

)

≡

p

n

+

1

x

-

p

n

x

and plot it for

x

∈

ℝ

.

It is clear that

A

G

⁢

(

0

,

n

)

=

0

.

For

x

<

0

,

A

G

⁢

(

x

,

n

)

is

negative

, and if

x

→

-

∞

then

A

G

⁢

(

x

,

n

)

→

-

∞

.

For

x

>

0

,

A

G

⁢

(

x

,

n

)

is positive, and if

x

→

+

∞

then

A

G

⁢

(

x

,

n

)

→

+

∞

.

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

≈

0.5671481302

⁢

…

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

⁢

…

Numerical plots for the first

2

×

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

→

∞

.

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

×

10

3

primes.

Plots for the first

2

×

10

4

primes.

Plots for the first

2

×

10

5

primes.

Plots for the first

2

×

10

6

primes.

Plots for the first

2

×

10

9

primes.

Plots for the first

2

×

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

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