- 1. Integers then Equals

Posted by: "Sebastian Martin Ruiz" s_m_ruiz@... s_m_ruiz

Date: Sat Mar 14, 2009 12:11 pm ((PDT))

Hello all:

Hello Sebastián

Let P(n)

the n-th prime number. Let Sqrt(P)=p^(1/2)

Kermit:

****

p1 = 2

p2 = 3

p3 = 5

2 + 7 = 9

5 - 1 = 4

sqrt(p1+7) = 3

sqrt(p3-1) = 2

****

If

Sqrt(P(n-1)+7) and Sqrt(P(n+1)-1) are both integers then:

Kermit:

*****

If sqrt(p(n-1) + 7) is an integer, then p(n-1) = 0 or 2 mod 3.

If sqrt(p(n+1) -1) is an integer, then p(n+1) = 1 or 2 mod 3.

p(n-1) = 0 mod 3 implies p = 3.

sqrt(3 + 7) is not an integer,

Thus

If sqrt(p(n-1) + 7) is an integer, then p(n-1) = 2 mod 3

if sqrt(p(n+1) -1) is an integer, then p(n+1) = 1 or 2 mod 3.

If sqrt(p(n-1) + 7) is an integer > 3,

and

sqrt(p(n+1) - 1) is an integer, then

p(n+1) = p(n-1) + 8

and p(n) might be p(n-1) + 2 or p(n-1) + 6.

sqrt(p1+7) = 3

sqrt(p3-1) = 2

*****

1)

Sqrt(P(n-1)+7)=Sqrt(P(n+1)-1)

Kermit:

****

Provided n is sufficiently large.

3**2 - 7 = 2

4**2 - 7 = 9

5**2 - 7 = 19

2,3,5,7,11,13,17,19

p8 = 19

p9 = 23

p10 = 29

sqrt(28) > sqrt(25)

******

2)

P(n)=P(n-1)+2 (Twin Primes)

Kermit:

***

I expect there to be exceptions to this also.

Because

If p(n+1) = p(n-1) + 8,

p(n) might be p(n-1) + 2,

or

p(n) might be p(n-1) + 6

******

3) P(n+1)=P(n-1)+8

Sincerely

Sebastián Martín Ruiz - --- In primenumbers@yahoogroups.com, "Maximilian Hasler" <maximilian.hasler@...> wrote:
>

O.k., put it in.

> > --- In primenumbers@yahoogroups.com, "Werner D. Sand" wrote:

> > >

> > > In words: If N is a square (N=m²) and N+1 and N-7 are primes,

> > > then N-5 is a prime, too.

> >

> > Stated like this, it is indeed more or less trivial.

>

> er... I read but you didn't write:

> "...and there is a prime between N-7 and N+1, ..."

>

> else we have counter-examples for m =

> 54,66,90,156,240,270,306,474,570,576,636,750,780,1080,1320,1350,2034,

> 2154,2406,2700,2760,3204,3240,3306,3480,3516,3756,3774,3984,4056,4086,

> 4140,4146,4176,4716,4734,4794,5154,5370,5424,5550,5664,5700,5850,5856,

> 5970,6030,6060,6120,6366,6576,6714,6786,7050,7164,8196,8424,8454,8940,

> 9180,9246,9486,9696,9804...

>