--- In

primenumbers@yahoogroups.com, "ianredwood" <ianredwood@...> wrote:

>

> Whoops - I've got to say that again.

> What's the highest known n for which, for any 7 <= k <= n, pi(k/2) > pi(k) - pi(k/2)?

> Is it bigger than 10^(60)?

>

pi(k/2) > pi(k) - pi(k/2)

<=> pi(k/2)*2 > pi(k)

The function pi(k/2)*2 - pi(k) tends to infinity

in view of the asymptotic behavious of the pi() function

so for all k>5 it is nonnegative and for all k>21 it is strictly positive.

vector(20,i,(primepi((1+i)\2)*2-primepi(1+i))) /* i+1 to avoid pi(0) */

= [-1, -2, 0, -1, 1, 0, 0, 0, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 0, 0]

vector(99,i,(primepi((1+i)\2)*2-primepi(1+i)))

= [-1, -2, 0, -1, 1, 0, 0, 0, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 0, 0, 2, 1, 1, 1, 3, 3, 3, 2, 2, 1, 1, 1, 3, 3, 3, 2, 4, 4, 4, 3, 3,

2, 2, 2, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 4, 3, 3, 2, 4, 4, 4, 4, 4, 3, 3, 3, 3, 2, 2, 1, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 3, 3, 5,

5, 5, 4, 4, 4, 4, 4, 6, 6, 6, 5, 5, 5, 5]

Maximilian