- Let P represent an odd prime>3 and N represent the number of primes from

and including 2 to and including P. Then in the set of positive integers

beginning with (2*P+N) and ending with 2*(P+N), there exists a prime Q

where the number of primes from and including 2 to and including Q is

equal to (2*N). The difference D=(Q-2*P) is always a number between N and

2*N.

For example, take P=47. N=15. (2*P+N)=109. 2*(P+N)=124. The (2*N)th prime

is 113. It lies between 109 and 124. D=(Q-2*P)=19. It lies between N=15

and 2*N =30.

Here is a neat consequence of the above statement. Take the expression

((4*P+3*N)/2), where P represents a prime>2 and N represents the number

of primes from and including 2 to and including P. Then for certain

values of P and its corresponding N, the expression will evaluate to a

prime Q, where the number of primes from and including 2 to and including

Q is equal to (2*N). The Q's are symmetrically located between (2*P+N)

and 2*(P+N). The D's are symmetrically located between N and 2*N.

There seems to be an inexhaustible supply of such P's. Here is a partial

list of the first 10 of such P's. (43, 163, 373, 397, 491, 1997, 2339,

4691, 7331, 12149), and the list of corresponding N's. (14, 38, 74, 78,

94, 302, 346, 634, 934, 1454).

The only web reference I was able to find for this, is the

Bertrand-Chebyshev theorem, which says that there is always at least one

prime between N and 2*N-2. I can see no connection. I have been trying to

prove that it is an obvious consequence of the prime number theorem, but

can't see my way clear. Can anyone help?

Thanks folks. Any comments would be appreciated.

Bill Sindelar

____________________________________________________________

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>

N and

> Let P represent an odd prime>3 and N represent the number of primes from

> and including 2 to and including P. Then in the set of positive integers

> beginning with (2*P+N) and ending with 2*(P+N), there exists a prime Q

> where the number of primes from and including 2 to and including Q is

> equal to (2*N). The difference D=(Q-2*P) is always a number between

> 2*N.

prime

> For example, take P=47. N=15. (2*P+N)=109. 2*(P+N)=124. The (2*N)th

> is 113. It lies between 109 and 124. D=(Q-2*P)=19. It lies between N=15

including

> and 2*N =30.

> Here is a neat consequence of the above statement. Take the expression

> ((4*P+3*N)/2), where P represents a prime>2 and N represents the number

> of primes from and including 2 to and including P. Then for certain

> values of P and its corresponding N, the expression will evaluate to a

> prime Q, where the number of primes from and including 2 to and

> Q is equal to (2*N). The Q's are symmetrically located between (2*P+N)

trying to

> and 2*(P+N). The D's are symmetrically located between N and 2*N.

> There seems to be an inexhaustible supply of such P's. Here is a partial

> list of the first 10 of such P's. (43, 163, 373, 397, 491, 1997, 2339,

> 4691, 7331, 12149), and the list of corresponding N's. (14, 38, 74, 78,

> 94, 302, 346, 634, 934, 1454).

> The only web reference I was able to find for this, is the

> Bertrand-Chebyshev theorem, which says that there is always at least one

> prime between N and 2*N-2. I can see no connection. I have been

> prove that it is an obvious consequence of the prime number theorem, but

Someone's probably privately replied already, but yes your result is a

> can't see my way clear. Can anyone help?

> Thanks folks. Any comments would be appreciated.

> Bill Sindelar

consequence of the prime number theorem (although it certainly doesn't

prove that your result will always hold):

The nth prime occurs around n*log(n), which is close to your p.

The (2*n)th prime occurs around 2*n*log(2*n).

But log(2*n) = log(2) + log(n), so the (2*n)th prime occurs around

2*n*log(n) + 2*n*log(2) =~ 2*n*log(n) + 1.386*n

But n*log(n) is close to your p, so the (2*n)th prime occurs around

2*p + 1.386*n, which is in between 2*p + n and 2*p + 2*n as you have

found.

Mark

. - Sindelar wrote in primenumbers message 19425:

< Take the expression ((4*P+3*N)/2), where P represents a prime>2 and N

represents the number of primes from and including 2 to and including P.

Then for certain values of P and its corresponding N, the expression will

evaluate to a prime Q, where the number of primes from and including 2 to

and including Q is equal to (2*N). The Q's are symmetrically located

between (2*P+N) and 2*(P+N). The D's are symmetrically located between N

and 2*N.

There seems to be an inexhaustible supply of such P's. Here is a partial

list of the first 10 of such P's. (43, 163, 373, 397, 491, 1997, 2339,

4691, 7331, 12149), and the list of corresponding N's. (14, 38, 74, 78,

94, 302, 346, 634, 934, 1454)...>

David Broadhurst wrote to Sindelar:

<<I believe that there is /no/ solution to

4*prime(n) + 3*n = 2*prime(2*n)

for n > 352314

Explanation:

1) the asymptotic value of

R(n) = (2*prime(2*n) - 4*prime(n))/n

is

R(infinity) = 4*log(2) = 2.7725887...

by the Prime Number Theorem.

2) For small n, there are excursions with R(n) >= 3,

but these soon cease, with

R(352314) = 3

being your last solution

and

R(352316) = 3 + 1/176158

being the last value of R(n) >= 3.

3) Note that you may use

http://primes.utm.edu/nthprime/

to find R(n) for 2*n <= 10^12.

For example:

The 500,000,000,000th prime is 14,638,944,639,703

The 1,000,000,000,000th prime is 29,996,224,275,833

showing that

R(5*10^11) = (2*29996224275833 - 4*14638944639703)/(5*10^11)

= 2.873339985708

is now significantly smaller than 3,

yet still some way above the asymptote 2.7725887...

David

PS: You may quote this argument in "primenumbers", if you wish. >>

Thank you David, for taking the time to respond to my post. It was quite

a surprise. I had not expected to see my paltry list of 10 N's expanded

to 92 so quickly, and to cap it off, be presented with an argument that

makes a case for there being a limit to the number of solutions to the

equation (4*P+3*N)=2*(the (2*N)th prime). Put another way, this means

that for any N>352314, the difference between the (2*N)th prime and twice

the Nth prime can never equal (3*N)/2.

I feel a bit uneasy about this, asymptotics can be tricky. I think the

idea would be of interest to many in the group and I hope it will provoke

a lively discussion. Best regards

Bill Sindelar

____________________________________________________________

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Kulsha's graphic. I've done a lot of calculating and believe everything

strongly suggests that the following statement may be true.

Let P represent the Nth prime and Q represent the (2*N)th prime.

Let K represent a rational number between the integers 1 and 2.

Then the equation (Q-2*P)=K*N always has a least one solution. The

maximum number of solutions possible is when K=3/2.

For those in the group that may be interested, here is David's reason for

believing that the equation (4*P+3*N)=2*(the (2*N)th prime), which is

equivalent to (Q-2*P)=K*N with K=(3/2), has a limited number of

solutions. (David calculated 92)> 1) First, I worked out the second term in the asymptotic

Thanks folks. Anyone care to comment?

> expansion of

>

> R(n) = (2*prime(2*n) - 4*prime(n))/n

>

> as here:

>

> R(n) ~ 4*log(2)*(1 + 1/log(n))

>

> 2) Then I proved, by explicit computation,

> that there is no solution to R(n)=3

> with 1500000 > n > 352314 (file attached)

>

> 3) In the course of this proof, I saw, by outputting

>

> [n, R(n), 4*log(2)*(1+1/log(n))]

>

> at intervals of 10,000

> that the actual value, R(n), oscillates only modestly

> about the asymptote 4*log(2)*(1+1/log(n))

> so I believe that I have cause to believe

> (yet cannot prove) that n=352314 is the last solution with

> R(n)=3.

>

> David

>

> PS: Feel free to share this, if you might like to...

Bill Sindelar

____________________________________________________________

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