- It's true that there is at least one prime between n^2 and (n+1)^2,

although I can't recall if it has been proven.

Now n*(n+1) lies between n^2 and (n+1)^2, and we observe a tighter

result, namely that for n>1 there is always a prime between

n^2 and n*(n+1)

and also between

n*(n+1) and (n+1)^2

(This is yet another way to divide all odd primes into two groups!)

******************

Also, I was toying with a minimum value of c such that there is

always a prime between n^c and (n+1)^c.

I'm not totally certain but pretty sure that the lowest value of c is

about 1.5683. Some lower values come very close but no cigar.

For example, there is one prime between 12^1.5683 and 13^1.5683. That

is, 53 lies between 49.258 and 55.846.

Let s(n) be the number of primes between n^1.5683 and (n+1)^1.5683.

s(n) for n=1 to n=14 is 1,2,1,1,1,2,1,2,1,1,2,1,2,1.

s(n) = 1 for 18 values of n from 1 to 100.

s(n) = 1 for 33 values of n from 1 to 737. After that s(n) is always

greater than 1.

Mark - On Monday 28 February 2005 17:29, you wrote:
> It's true that there is at least one prime between n^2 and (n+1)^2,

No, it hasn't.

> although I can't recall if it has been proven.

>

If this result is proved, then the above follows immediately. So it hasn't

> Now n*(n+1) lies between n^2 and (n+1)^2, and we observe a tighter

> result, namely that for n>1 there is always a prime between

>

> n^2 and n*(n+1)

>

> and also between

>

> n*(n+1) and (n+1)^2

>

> (This is yet another way to divide all odd primes into two groups!)

been proven either.

>

By the PNT, the n-th prime is ~n log n, so it follows that all c's are

> ******************

>

>

> Also, I was toying with a minimum value of c such that there is

> always a prime between n^c and (n+1)^c.

>

> I'm not totally certain but pretty sure that the lowest value of c is

> about 1.5683. Some lower values come very close but no cigar.

>

>

> For example, there is one prime between 12^1.5683 and 13^1.5683. That

> is, 53 lies between 49.258 and 55.846.

acceptable at infinity, since any power > 1 of n grows faster than n log n.

Whatever value of c that ends up being the smallest possible is just a quirk

of the start of the sequence. If you disregard the `first' primes, then any c

= 1 + e will do (for a suitable definition of `first' that's a function of

e).

> Let s(n) be the number of primes between n^1.5683 and (n+1)^1.5683.

See the PNT argument above.

>

>

> s(n) for n=1 to n=14 is 1,2,1,1,1,2,1,2,1,1,2,1,2,1.

>

> s(n) = 1 for 18 values of n from 1 to 100.

>

> s(n) = 1 for 33 values of n from 1 to 737. After that s(n) is always

> greater than 1.

Décio

[Non-text portions of this message have been removed] - "Erdos proved that there exist at least one prime of the form 4k+1

and at least one prime of the form 4k+3 between n and 2n for all n >

6."

This was stated at:

Eric W. Weisstein. "Modular Prime Counting Function." From MathWorld-

-A Wolfram Web Resource.

http://mathworld.wolfram.com/ModularPrimeCountingFunction.html

Does anyone know where is this proof is?

Second, how does this NOT prove Mark's question?

I mean when can with n > 6 and m^2 >6,

n < p1 < m^2 < (m+1)^2 = m^2 + 2*m + 1 < p2 < 2*n,

happen without a prime inside p1 < m^2 < (m+1)^2 = m^2 + 2*m + 1 <

p2?

Note n > 6 and m^2 >6:

If m^2 < n < 2*n <(m+1)^2 then a primes exist inside the squares by

Erdos's and by Bertrand's.

If n < m^2 < 2*n < (m+1)^2 and both Erdos primes are n < p1 < p2 <

m^2 then a prime exist inside the squares by at least Bertrand's

Postulate for prime p3.

If n < m^2 < 2*n < (m+1)^2 and a Erdos prime is m^2 < p1 < p2 < 2*n

or p1 < m^2 < p2 < 2*n then a prime exist inside the squares.

If m^2 < n < (m+1)^2 < 2*n and a Erdos prime is m^2 < p1 < p2 < 2*n

or m^2 < p1 < 2*n < p2 then a prime exist inside the squares.

If m^2 < n <(m+1)^2 < 2*n and both prime are between (m+1)^2 and 2*n

then Bertrand's Postulate holds.

If m < (m+1)^2 < n < 2*n or n < 2*n < m < (m+1)^2, then we have bad

unrelated sets in this examination, But a prime p1 < m^2 or (m+1)^2

< p2 will be near enough to m^2 or (m+1)^2 so that p3 can be a

solution by Bertrand's (or by Bertrand's used multiple times).

Note that n*(n+1) can be subtituded in the logic above because of

the two nearby primes.

As for the also part, Mark see the these pages:

http://mathworld.wolfram.com/AndricasConjecture.html

http://mathworld.wolfram.com/OmegaConstant.html

Look at the generalization of Andrica's conjecture considers the

equation

p_(n+1)^x - p_(n)^x = 1

I know that the sign has been changed for your case, but it may give

you insight.

Nick

--- In primenumbers@yahoogroups.com, "Mark Underwood"

<mark.underwood@s...> wrote:>

^2,

>

> It's true that there is at least one prime between n^2 and (n+1)

> although I can't recall if it has been proven.

is

>

>

> Now n*(n+1) lies between n^2 and (n+1)^2, and we observe a tighter

> result, namely that for n>1 there is always a prime between

>

> n^2 and n*(n+1)

>

> and also between

>

> n*(n+1) and (n+1)^2

>

> (This is yet another way to divide all odd primes into two groups!)

>

>

> ******************

>

>

> Also, I was toying with a minimum value of c such that there is

> always a prime between n^c and (n+1)^c.

>

> I'm not totally certain but pretty sure that the lowest value of c

> about 1.5683. Some lower values come very close but no cigar.

That

>

>

> For example, there is one prime between 12^1.5683 and 13^1.5683.

> is, 53 lies between 49.258 and 55.846.

^1.5683.

>

>

> Let s(n) be the number of primes between n^1.5683 and (n+1)

>

always

>

> s(n) for n=1 to n=14 is 1,2,1,1,1,2,1,2,1,1,2,1,2,1.

>

> s(n) = 1 for 18 values of n from 1 to 100.

>

> s(n) = 1 for 33 values of n from 1 to 737. After that s(n) is

> greater than 1.

>

>

> Mark - The Erdos proof is available at Ask Dr. Math - not that I understand it!

Bob

nick_honolson <nick_honolson@...> wrote:

"Erdos proved that there exist at least one prime of the form 4k+1

and at least one prime of the form 4k+3 between n and 2n for all n >

6."

This was stated at:

Eric W. Weisstein. "Modular Prime Counting Function." From MathWorld-

-A Wolfram Web Resource.

http://mathworld.wolfram.com/ModularPrimeCountingFunction.html

Does anyone know where is this proof is?

Second, how does this NOT prove Mark's question?

I mean when can with n > 6 and m^2 >6,

n < p1 < m^2 < (m+1)^2 = m^2 + 2*m + 1 < p2 < 2*n,

happen without a prime inside p1 < m^2 < (m+1)^2 = m^2 + 2*m + 1 <

p2?

Note n > 6 and m^2 >6:

If m^2 < n < 2*n <(m+1)^2 then a primes exist inside the squares by

Erdos's and by Bertrand's.

If n < m^2 < 2*n < (m+1)^2 and both Erdos primes are n < p1 < p2 <

m^2 then a prime exist inside the squares by at least Bertrand's

Postulate for prime p3.

If n < m^2 < 2*n < (m+1)^2 and a Erdos prime is m^2 < p1 < p2 < 2*n

or p1 < m^2 < p2 < 2*n then a prime exist inside the squares.

If m^2 < n < (m+1)^2 < 2*n and a Erdos prime is m^2 < p1 < p2 < 2*n

or m^2 < p1 < 2*n < p2 then a prime exist inside the squares.

If m^2 < n <(m+1)^2 < 2*n and both prime are between (m+1)^2 and 2*n

then Bertrand's Postulate holds.

If m < (m+1)^2 < n < 2*n or n < 2*n < m < (m+1)^2, then we have bad

unrelated sets in this examination, But a prime p1 < m^2 or (m+1)^2

< p2 will be near enough to m^2 or (m+1)^2 so that p3 can be a

solution by Bertrand's (or by Bertrand's used multiple times).

Note that n*(n+1) can be subtituded in the logic above because of

the two nearby primes.

As for the also part, Mark see the these pages:

http://mathworld.wolfram.com/AndricasConjecture.html

http://mathworld.wolfram.com/OmegaConstant.html

Look at the generalization of Andrica's conjecture considers the

equation

p_(n+1)^x - p_(n)^x = 1

I know that the sign has been changed for your case, but it may give

you insight.

Nick

--- In primenumbers@yahoogroups.com, "Mark Underwood"

<mark.underwood@s...> wrote:>

^2,

>

> It's true that there is at least one prime between n^2 and (n+1)

> although I can't recall if it has been proven.

is

>

>

> Now n*(n+1) lies between n^2 and (n+1)^2, and we observe a tighter

> result, namely that for n>1 there is always a prime between

>

> n^2 and n*(n+1)

>

> and also between

>

> n*(n+1) and (n+1)^2

>

> (This is yet another way to divide all odd primes into two groups!)

>

>

> ******************

>

>

> Also, I was toying with a minimum value of c such that there is

> always a prime between n^c and (n+1)^c.

>

> I'm not totally certain but pretty sure that the lowest value of c

> about 1.5683. Some lower values come very close but no cigar.

That

>

>

> For example, there is one prime between 12^1.5683 and 13^1.5683.

> is, 53 lies between 49.258 and 55.846.

^1.5683.

>

>

> Let s(n) be the number of primes between n^1.5683 and (n+1)

>

always

>

> s(n) for n=1 to n=14 is 1,2,1,1,1,2,1,2,1,1,2,1,2,1.

>

> s(n) = 1 for 18 values of n from 1 to 100.

>

> s(n) = 1 for 33 values of n from 1 to 737. After that s(n) is

> greater than 1.

Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

>

>

> Mark

The Prime Pages : http://www.primepages.org/

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[Non-text portions of this message have been removed] - Thank you Nick, that was helpful! Erdos figured prominently as an

amazing 'networker' in the book I recently read called The Music of

the Primes. And that Wolfram site is amazing.

Mike, I see now that even lesser minds can think alike to greater

minds! One wonders how much of this stuff has been hashed through

even hundreds of years go. Oh well the joy is in the (re)discovery.

You are waxing philosophical when you say

"There is no special "reason" preventing more primes from turning up -

it's just the way the cookie crumbles sometimes."

Perhaps, yet maybe *God* knows the 'reason' why cookies crumble

exactly the way they do :-)

Mark

--- In primenumbers@yahoogroups.com, "nick_honolson"

<nick_honolson@y...> wrote:>

MathWorld-

> "Erdos proved that there exist at least one prime of the form 4k+1

> and at least one prime of the form 4k+3 between n and 2n for all n

>

> 6."

> This was stated at:

> Eric W. Weisstein. "Modular Prime Counting Function." From

> -A Wolfram Web Resource.

by

> http://mathworld.wolfram.com/ModularPrimeCountingFunction.html

>

> Does anyone know where is this proof is?

> Second, how does this NOT prove Mark's question?

>

> I mean when can with n > 6 and m^2 >6,

> n < p1 < m^2 < (m+1)^2 = m^2 + 2*m + 1 < p2 < 2*n,

> happen without a prime inside p1 < m^2 < (m+1)^2 = m^2 + 2*m + 1 <

> p2?

>

> Note n > 6 and m^2 >6:

>

> If m^2 < n < 2*n <(m+1)^2 then a primes exist inside the squares

> Erdos's and by Bertrand's.

2*n

>

> If n < m^2 < 2*n < (m+1)^2 and both Erdos primes are n < p1 < p2 <

> m^2 then a prime exist inside the squares by at least Bertrand's

> Postulate for prime p3.

>

> If n < m^2 < 2*n < (m+1)^2 and a Erdos prime is m^2 < p1 < p2 <

> or p1 < m^2 < p2 < 2*n then a prime exist inside the squares.

2*n

>

> If m^2 < n < (m+1)^2 < 2*n and a Erdos prime is m^2 < p1 < p2 <

> or m^2 < p1 < 2*n < p2 then a prime exist inside the squares.

2*n

>

> If m^2 < n <(m+1)^2 < 2*n and both prime are between (m+1)^2 and

> then Bertrand's Postulate holds.

bad

>

> If m < (m+1)^2 < n < 2*n or n < 2*n < m < (m+1)^2, then we have

> unrelated sets in this examination, But a prime p1 < m^2 or (m+1)^2

give

> < p2 will be near enough to m^2 or (m+1)^2 so that p3 can be a

> solution by Bertrand's (or by Bertrand's used multiple times).

>

> Note that n*(n+1) can be subtituded in the logic above because of

> the two nearby primes.

>

>

> As for the also part, Mark see the these pages:

>

> http://mathworld.wolfram.com/AndricasConjecture.html

> http://mathworld.wolfram.com/OmegaConstant.html

>

> Look at the generalization of Andrica's conjecture considers the

> equation

>

> p_(n+1)^x - p_(n)^x = 1

>

> I know that the sign has been changed for your case, but it may

> you insight.

tighter

>

> Nick

>

>

> --- In primenumbers@yahoogroups.com, "Mark Underwood"

> <mark.underwood@s...> wrote:

> >

> >

> > It's true that there is at least one prime between n^2 and (n+1)

> ^2,

> > although I can't recall if it has been proven.

> >

> >

> > Now n*(n+1) lies between n^2 and (n+1)^2, and we observe a

> > result, namely that for n>1 there is always a prime between

groups!)

> >

> > n^2 and n*(n+1)

> >

> > and also between

> >

> > n*(n+1) and (n+1)^2

> >

> > (This is yet another way to divide all odd primes into two

> >

c

> >

> > ******************

> >

> >

> > Also, I was toying with a minimum value of c such that there is

> > always a prime between n^c and (n+1)^c.

> >

> > I'm not totally certain but pretty sure that the lowest value of

> is

> > about 1.5683. Some lower values come very close but no cigar.

> >

> >

> > For example, there is one prime between 12^1.5683 and 13^1.5683.

> That

> > is, 53 lies between 49.258 and 55.846.

> >

> >

> > Let s(n) be the number of primes between n^1.5683 and (n+1)

> ^1.5683.

> >

> >

> > s(n) for n=1 to n=14 is 1,2,1,1,1,2,1,2,1,1,2,1,2,1.

> >

> > s(n) = 1 for 18 values of n from 1 to 100.

> >

> > s(n) = 1 for 33 values of n from 1 to 737. After that s(n) is

> always

> > greater than 1.

> >

> >

> > Mark