- pi(x) - pi(x/7) > sqrt(x) proved by N. Shapiro

pi(x) > sqrt(x) + pi(x/7)

let x = p_(n +1) and p_n so that

pi( p_(n +1)) - pi( p_n) > sqrt( p_(n +1)) + pi( p_(n +1) /7) - (sqrt( p_n) +

pi( p_n /7))

or

n+1 - n > sqrt( p_(n +1)) - sqrt( p_n) + pi( p_(n +1) /7) - pi( p_n /7)

1 > sqrt( p_(n +1)) - sqrt( p_n)

QED

Does anyone see a problem here?

John Nicholson > pi(x) > sqrt(x) + pi(x/7)

...

> pi( p_(n +1)) - pi( p_n) >

...

> sqrt( p_(n +1)) + pi( p_(n +1) /7) - (sqrt( p_n) + pi( p_n /7))

> QED

Yes, I do. While 10 is greater than 6 and 8 is greater than 3, (10-8) is

> Does anyone see a problem here?

not greater than (6-3).

Peter- --- In primenumbers@yahoogroups.com,

"John W. Nicholson" <reddwarf2956@...> wrote:

> pi(x) - pi(x/7) > sqrt(x) proved by N. Shapiro

A proof, for integer x > 2, was given by Harold Shapiro

in 1953 and may be found here:

http://tinyurl.com/3yycvms

The inequality is true by inspection for 2 < x < 10590

and for larger x follows from a stronger result by

Ramanujan, given in Eq. (3.2) of Shapiro's paper.

David - David thanks,

Sorry I left off the "Harold" in Harold N. Shapiro.

John W. Nicholson

--- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:

>

>

>

> --- In primenumbers@yahoogroups.com,

> "John W. Nicholson" <reddwarf2956@> wrote:

>

> > pi(x) - pi(x/7) > sqrt(x) proved by N. Shapiro

>

> A proof, for integer x > 2, was given by Harold Shapiro

> in 1953 and may be found here:

>

> http://tinyurl.com/3yycvms

>

> The inequality is true by inspection for 2 < x < 10590

> and for larger x follows from a stronger result by

> Ramanujan, given in Eq. (3.2) of Shapiro's paper.

>

> David

>