## Re: [PrimeNumbers] pi(x) - pi(x/7) > sqrt(x)

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• ... Yes, I do. While 10 is greater than 6 and 8 is greater than 3, (10-8) is not greater than (6-3). Peter
Message 1 of 4 , Jul 14, 2010
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> 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
> Does anyone see a problem here?

Yes, I do. While 10 is greater than 6 and 8 is greater than 3, (10-8) is
not greater than (6-3).

Peter
• ... 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
Message 2 of 4 , Jul 14, 2010
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"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
Message 3 of 4 , Jul 14, 2010
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David thanks,

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

John W. Nicholson

>
>
>
> "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
>
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