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Strengthening Bertram's postulate using the prime number theorem
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In its page on Bertram's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
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In its page on Bertram's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
1
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
– fleablood
Jul 16 at 0:00
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
– Ethan Bolker
Jul 16 at 0:04
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In its page on Bertram's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
In its page on Bertram's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
asked Jul 15 at 23:54
Ethan Bolker
35.8k54299
35.8k54299
1
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
– fleablood
Jul 16 at 0:00
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
– Ethan Bolker
Jul 16 at 0:04
add a comment |Â
1
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
– fleablood
Jul 16 at 0:00
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
– Ethan Bolker
Jul 16 at 0:04
1
1
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
– fleablood
Jul 16 at 0:00
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
– fleablood
Jul 16 at 0:00
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
– Ethan Bolker
Jul 16 at 0:04
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
– Ethan Bolker
Jul 16 at 0:04
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ displaystyle varepsilon >0$ there is a $displaystyle n_0>0 $ such that for all $displaystyle n>n_0 $ there is a prime $ displaystyle p$ such $displaystyle n<p<(1+varepsilon )n$. It can be shown, for instance,
that
$displaystyle lim _nto infty frac pi ((1+varepsilon )n)-pi
> (n)n/log n=varepsilon , $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 at 0:12
fleablood
60.5k22575
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
add a comment |Â
up vote
1
down vote
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac125 log^2 x right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^18,$ we get gap $g < log^2 p.$
answered Jul 16 at 0:22
Will Jagy
97.2k594196
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ displaystyle varepsilon >0$ there is a $displaystyle n_0>0 $ such that for all $displaystyle n>n_0 $ there is a prime $ displaystyle p$ such $displaystyle n<p<(1+varepsilon )n$. It can be shown, for instance,
that
$displaystyle lim _nto infty frac pi ((1+varepsilon )n)-pi
> (n)n/log n=varepsilon , $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 at 0:12
fleablood
60.5k22575
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
add a comment |Â
up vote
1
down vote
accepted
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ displaystyle varepsilon >0$ there is a $displaystyle n_0>0 $ such that for all $displaystyle n>n_0 $ there is a prime $ displaystyle p$ such $displaystyle n<p<(1+varepsilon )n$. It can be shown, for instance,
that
$displaystyle lim _nto infty frac pi ((1+varepsilon )n)-pi
> (n)n/log n=varepsilon , $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 at 0:12
fleablood
60.5k22575
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ displaystyle varepsilon >0$ there is a $displaystyle n_0>0 $ such that for all $displaystyle n>n_0 $ there is a prime $ displaystyle p$ such $displaystyle n<p<(1+varepsilon )n$. It can be shown, for instance,
that
$displaystyle lim _nto infty frac pi ((1+varepsilon )n)-pi
> (n)n/log n=varepsilon , $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 at 0:12
fleablood
60.5k22575
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ displaystyle varepsilon >0$ there is a $displaystyle n_0>0 $ such that for all $displaystyle n>n_0 $ there is a prime $ displaystyle p$ such $displaystyle n<p<(1+varepsilon )n$. It can be shown, for instance,
that
$displaystyle lim _nto infty frac pi ((1+varepsilon )n)-pi
> (n)n/log n=varepsilon , $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 at 0:12
fleablood
60.5k22575
edited Jul 16 at 2:28
answered Jul 16 at 0:12
fleablood
60.5k22575
answered Jul 16 at 0:12
fleablood
60.5k22575
answered Jul 16 at 0:12
fleablood
60.5k22575
60.5k22575
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
add a comment |Â
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
– Steve Kass
Jul 16 at 0:34
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
– fleablood
Jul 16 at 2:19
add a comment |Â
up vote
1
down vote
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac125 log^2 x right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^18,$ we get gap $g < log^2 p.$
answered Jul 16 at 0:22
Will Jagy
97.2k594196
add a comment |Â
up vote
1
down vote
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac125 log^2 x right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^18,$ we get gap $g < log^2 p.$
answered Jul 16 at 0:22
Will Jagy
97.2k594196
add a comment |Â
up vote
1
down vote
up vote
1
down vote
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac125 log^2 x right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^18,$ we get gap $g < log^2 p.$
answered Jul 16 at 0:22
Will Jagy
97.2k594196
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac125 log^2 x right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^18,$ we get gap $g < log^2 p.$
answered Jul 16 at 0:22
Will Jagy
97.2k594196
edited Jul 16 at 0:38
answered Jul 16 at 0:22
Will Jagy
97.2k594196
answered Jul 16 at 0:22
Will Jagy
97.2k594196
answered Jul 16 at 0:22
Will Jagy
97.2k594196
97.2k594196
add a comment |Â
add a comment |Â
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1
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
– fleablood
Jul 16 at 0:00
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
– Ethan Bolker
Jul 16 at 0:04