## Correction on conjecture.

Expand Messages
• Kermit Rose wrote: This is correction of my previous post. I mistakenly said an upper bound on the number of primes between (n+1) and (2 n) when I mean to say
Message 1 of 2 , Sep 5, 2009
• 0 Attachment
Kermit Rose wrote:

This is correction of my previous post.

I mistakenly said an upper bound on the number of primes between (n+1)
and (2 n)
when I mean to say a lower bound on the number of primes between (n+1)
and (2 n).

>
> In order to get an intuitive notion of the number
> of primes between (n+1) and (2n) for large n,
> I considered the following.
>
> As a result of these observations I made a
> numerical difference ,
> number of primes between 1 and n,
> minus
> the number of primes between (n+1) and (2n).
>
>
> Between 1 and 5 there are 2 even integers.
> Between 6 and 10 there are 3 even integers.
>
> Between 1 and 6 there are 3 even integers.
> Between 7 and 12 there are 3 even integers.
>
> Between 1 and 7 there are 3 even integers.
> Between 8 and 14 there are 4 even integers.
>
> Between 1 and 15 there are 5 multiplies of 3.
> Between 16 and 30 there are 5 multiplies of 3.
>
> Between 1 and 16 there are 5 multiplies of 3.
> Between 17 and 32 there are 5 multiplies of 3.
>
> Between 1 and 17 there are 5 multiplies of 3.
> Between 18 and 34 there are 6 multiplies of 3.
>
> Between 1 and 18 there are 6 multiplies of 3.
> Between 19 and 36 there are 6 multiplies of 3.
>
>
> Between 1 and 15 there are 3 multiplies of 5.
> Between 16 and 30 there are 3 multiplies of 5.
>
> Between 1 and 16 there are 3 multiplies of 5.
> Between 17 and 32 there are 3 multiplies of 5.
>
> Between 1 and 17 there are 3 multiplies of 5.
> Between 18 and 34 there are 3 multiplies of 5.
>
> Between 1 and 18 there are 3 multiplies of 5.
> Between 19 and 36 there are 4 multiplies of 5.
>
>
> Speculation:
>
> The number of primes between 1 and n
> minus the number of primes between (n+1) and (2n)
>
> is LESS than or equal to the number of primes p,
> 1 < p < n such that
>
> n mod p > p/2.
>
>
>

The rational for this conjecture is that

for p < n, if n mod p is > p/2,

then there is one more multiple of p between n +1and 2n than between 1
and n.

There cannot be more than one extra multiple of p between n+1 and 2 n

This means that there is at most one extra composite number due to p
between
n+1 and n/2.

This means that there is 1 or zero less primes due to p, between n+1 and
(2n)

Thus there are two factors limiting the loss of primes between n+1 and (2n).

(1) The number of primes such that n mod p < p/2

(2) Each composite number between n+1 and (2n) is the product of 2 or
more primes.

Kermit
Your message has been successfully submitted and would be delivered to recipients shortly.