How to see that the prime gaps functions isn't eventually monotonic?
$begingroup$
Let $g(n)$ be the distance between the $n$th prime and the next.
By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic.
Further we know that it isn't eventually monotonic (meaning $g(k) le g(k+1) le g(k+2) le cdots$) by extremely advanced theorems like Zhangs result that some finite gap occurs infinitely often.
It was hinted to me that there is an easier proof though, that doesn't depend extremely advanced results. Could anyone point me to one please?
number-theory reference-request prime-gaps
$endgroup$
add a comment |
$begingroup$
Let $g(n)$ be the distance between the $n$th prime and the next.
By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic.
Further we know that it isn't eventually monotonic (meaning $g(k) le g(k+1) le g(k+2) le cdots$) by extremely advanced theorems like Zhangs result that some finite gap occurs infinitely often.
It was hinted to me that there is an easier proof though, that doesn't depend extremely advanced results. Could anyone point me to one please?
number-theory reference-request prime-gaps
$endgroup$
2
$begingroup$
Presumably, you mean "not eventually monotonic." Because it can clearly be demonstrated to not be monotonic. Such as $13-11<11-7$.
$endgroup$
– Thomas Andrews
Feb 10 '16 at 15:26
$begingroup$
How about that $[p#, p#+p]$ are all composite. Then all you need is that some gap is less than $p$
$endgroup$
– user334732
Dec 24 '18 at 23:38
add a comment |
$begingroup$
Let $g(n)$ be the distance between the $n$th prime and the next.
By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic.
Further we know that it isn't eventually monotonic (meaning $g(k) le g(k+1) le g(k+2) le cdots$) by extremely advanced theorems like Zhangs result that some finite gap occurs infinitely often.
It was hinted to me that there is an easier proof though, that doesn't depend extremely advanced results. Could anyone point me to one please?
number-theory reference-request prime-gaps
$endgroup$
Let $g(n)$ be the distance between the $n$th prime and the next.
By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic.
Further we know that it isn't eventually monotonic (meaning $g(k) le g(k+1) le g(k+2) le cdots$) by extremely advanced theorems like Zhangs result that some finite gap occurs infinitely often.
It was hinted to me that there is an easier proof though, that doesn't depend extremely advanced results. Could anyone point me to one please?
number-theory reference-request prime-gaps
number-theory reference-request prime-gaps
edited Dec 24 '18 at 23:39
user334732
4,28411240
4,28411240
asked Feb 10 '16 at 15:21
Brennan.TobiasBrennan.Tobias
651316
651316
2
$begingroup$
Presumably, you mean "not eventually monotonic." Because it can clearly be demonstrated to not be monotonic. Such as $13-11<11-7$.
$endgroup$
– Thomas Andrews
Feb 10 '16 at 15:26
$begingroup$
How about that $[p#, p#+p]$ are all composite. Then all you need is that some gap is less than $p$
$endgroup$
– user334732
Dec 24 '18 at 23:38
add a comment |
2
$begingroup$
Presumably, you mean "not eventually monotonic." Because it can clearly be demonstrated to not be monotonic. Such as $13-11<11-7$.
$endgroup$
– Thomas Andrews
Feb 10 '16 at 15:26
$begingroup$
How about that $[p#, p#+p]$ are all composite. Then all you need is that some gap is less than $p$
$endgroup$
– user334732
Dec 24 '18 at 23:38
2
2
$begingroup$
Presumably, you mean "not eventually monotonic." Because it can clearly be demonstrated to not be monotonic. Such as $13-11<11-7$.
$endgroup$
– Thomas Andrews
Feb 10 '16 at 15:26
$begingroup$
Presumably, you mean "not eventually monotonic." Because it can clearly be demonstrated to not be monotonic. Such as $13-11<11-7$.
$endgroup$
– Thomas Andrews
Feb 10 '16 at 15:26
$begingroup$
How about that $[p#, p#+p]$ are all composite. Then all you need is that some gap is less than $p$
$endgroup$
– user334732
Dec 24 '18 at 23:38
$begingroup$
How about that $[p#, p#+p]$ are all composite. Then all you need is that some gap is less than $p$
$endgroup$
– user334732
Dec 24 '18 at 23:38
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In brief: You can tweak the linked to argument. All you need in addition is a reasonable bound on the number of consecutive primes that have the same difference. It is not hard to show a bound linear in the difference.
Then proceed as in the linked question, that is show that this leads to an upper bound on the number of primes below a certain threshold that contradicts well-known results.
There it is that the number of primes below $N^2$ is at most linear in $N$. Here it will be that the number of primes below $N^2$ is at most quadratic in $N$. Both are absurd.
In detail:
Let us first show that the number of consecutive primes with difference $d$ is at most $2d$.
A difference $d$ is fixed. Let $p$ be a an auxiliary prime that does not divide $d$, for example the next prime after $d$ (which by Bertrand's postulate is $<2d$).
We show that there can be at most $p+2$ consecutive primes with difference $d$.
Let us consider $p_0, p_0 + d , p_0 + 2d , dots, p_0 + p d, p_0 + (p+1)d$. modulo $p$. We show at least one of these numbers is not prime.
Since $p nmid d$, we get that $p_0, p_0 + d , p_0 + 2d , dots,p_0 + (p-1)d$ covers all congruence classes modulo $p$. Thus in particular one of these is $0$ modulo $p$. This means it is not a prime unless it is equal to $p$ itself. However this is only possible for $p_0$ or $p_0 + d$, as $p < 2d$. In this case, $p_0 + pd$ or $p_0 + (p+1)d$, respectively, is divisible by $p$ too and thus not prime.
Thus we have shown that the difference $d$ can occur at most $p+1$ times successively, and thus if the sequence of differences is not decreasing it can occur at most $p+1$ times in total. Recall that $p+1 le 2d$. So, the difference $d$ occurs at most $2d$ times.
This implies that the size of the $1+ sum_{d=1}^D 2d$-th prime is at least
$2+ sum_{d=1}^D (2d)d$.
Now the former sum is $1 + D(D+1)$ and the latter sum is $2 + frac{D(D+1)(2D+1)}{3}$. This later expression is at least of size $D^3/2$ (leaving very small $D$ aside).
Yet by the prime number theorem we know that there are about $D^3/2/log (D^3/2)$ primes below $D^3/2$. For sufficiently large $D$ this is much larger than $1 + D(D+1)$, and thus yields a contradiction to the $1 + D(D+1)$-th prime being greater than $D^3/2$.
$endgroup$
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
1
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
|
show 7 more comments
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1 Answer
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1 Answer
1
active
oldest
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oldest
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active
oldest
votes
$begingroup$
In brief: You can tweak the linked to argument. All you need in addition is a reasonable bound on the number of consecutive primes that have the same difference. It is not hard to show a bound linear in the difference.
Then proceed as in the linked question, that is show that this leads to an upper bound on the number of primes below a certain threshold that contradicts well-known results.
There it is that the number of primes below $N^2$ is at most linear in $N$. Here it will be that the number of primes below $N^2$ is at most quadratic in $N$. Both are absurd.
In detail:
Let us first show that the number of consecutive primes with difference $d$ is at most $2d$.
A difference $d$ is fixed. Let $p$ be a an auxiliary prime that does not divide $d$, for example the next prime after $d$ (which by Bertrand's postulate is $<2d$).
We show that there can be at most $p+2$ consecutive primes with difference $d$.
Let us consider $p_0, p_0 + d , p_0 + 2d , dots, p_0 + p d, p_0 + (p+1)d$. modulo $p$. We show at least one of these numbers is not prime.
Since $p nmid d$, we get that $p_0, p_0 + d , p_0 + 2d , dots,p_0 + (p-1)d$ covers all congruence classes modulo $p$. Thus in particular one of these is $0$ modulo $p$. This means it is not a prime unless it is equal to $p$ itself. However this is only possible for $p_0$ or $p_0 + d$, as $p < 2d$. In this case, $p_0 + pd$ or $p_0 + (p+1)d$, respectively, is divisible by $p$ too and thus not prime.
Thus we have shown that the difference $d$ can occur at most $p+1$ times successively, and thus if the sequence of differences is not decreasing it can occur at most $p+1$ times in total. Recall that $p+1 le 2d$. So, the difference $d$ occurs at most $2d$ times.
This implies that the size of the $1+ sum_{d=1}^D 2d$-th prime is at least
$2+ sum_{d=1}^D (2d)d$.
Now the former sum is $1 + D(D+1)$ and the latter sum is $2 + frac{D(D+1)(2D+1)}{3}$. This later expression is at least of size $D^3/2$ (leaving very small $D$ aside).
Yet by the prime number theorem we know that there are about $D^3/2/log (D^3/2)$ primes below $D^3/2$. For sufficiently large $D$ this is much larger than $1 + D(D+1)$, and thus yields a contradiction to the $1 + D(D+1)$-th prime being greater than $D^3/2$.
$endgroup$
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
1
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
|
show 7 more comments
$begingroup$
In brief: You can tweak the linked to argument. All you need in addition is a reasonable bound on the number of consecutive primes that have the same difference. It is not hard to show a bound linear in the difference.
Then proceed as in the linked question, that is show that this leads to an upper bound on the number of primes below a certain threshold that contradicts well-known results.
There it is that the number of primes below $N^2$ is at most linear in $N$. Here it will be that the number of primes below $N^2$ is at most quadratic in $N$. Both are absurd.
In detail:
Let us first show that the number of consecutive primes with difference $d$ is at most $2d$.
A difference $d$ is fixed. Let $p$ be a an auxiliary prime that does not divide $d$, for example the next prime after $d$ (which by Bertrand's postulate is $<2d$).
We show that there can be at most $p+2$ consecutive primes with difference $d$.
Let us consider $p_0, p_0 + d , p_0 + 2d , dots, p_0 + p d, p_0 + (p+1)d$. modulo $p$. We show at least one of these numbers is not prime.
Since $p nmid d$, we get that $p_0, p_0 + d , p_0 + 2d , dots,p_0 + (p-1)d$ covers all congruence classes modulo $p$. Thus in particular one of these is $0$ modulo $p$. This means it is not a prime unless it is equal to $p$ itself. However this is only possible for $p_0$ or $p_0 + d$, as $p < 2d$. In this case, $p_0 + pd$ or $p_0 + (p+1)d$, respectively, is divisible by $p$ too and thus not prime.
Thus we have shown that the difference $d$ can occur at most $p+1$ times successively, and thus if the sequence of differences is not decreasing it can occur at most $p+1$ times in total. Recall that $p+1 le 2d$. So, the difference $d$ occurs at most $2d$ times.
This implies that the size of the $1+ sum_{d=1}^D 2d$-th prime is at least
$2+ sum_{d=1}^D (2d)d$.
Now the former sum is $1 + D(D+1)$ and the latter sum is $2 + frac{D(D+1)(2D+1)}{3}$. This later expression is at least of size $D^3/2$ (leaving very small $D$ aside).
Yet by the prime number theorem we know that there are about $D^3/2/log (D^3/2)$ primes below $D^3/2$. For sufficiently large $D$ this is much larger than $1 + D(D+1)$, and thus yields a contradiction to the $1 + D(D+1)$-th prime being greater than $D^3/2$.
$endgroup$
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
1
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
|
show 7 more comments
$begingroup$
In brief: You can tweak the linked to argument. All you need in addition is a reasonable bound on the number of consecutive primes that have the same difference. It is not hard to show a bound linear in the difference.
Then proceed as in the linked question, that is show that this leads to an upper bound on the number of primes below a certain threshold that contradicts well-known results.
There it is that the number of primes below $N^2$ is at most linear in $N$. Here it will be that the number of primes below $N^2$ is at most quadratic in $N$. Both are absurd.
In detail:
Let us first show that the number of consecutive primes with difference $d$ is at most $2d$.
A difference $d$ is fixed. Let $p$ be a an auxiliary prime that does not divide $d$, for example the next prime after $d$ (which by Bertrand's postulate is $<2d$).
We show that there can be at most $p+2$ consecutive primes with difference $d$.
Let us consider $p_0, p_0 + d , p_0 + 2d , dots, p_0 + p d, p_0 + (p+1)d$. modulo $p$. We show at least one of these numbers is not prime.
Since $p nmid d$, we get that $p_0, p_0 + d , p_0 + 2d , dots,p_0 + (p-1)d$ covers all congruence classes modulo $p$. Thus in particular one of these is $0$ modulo $p$. This means it is not a prime unless it is equal to $p$ itself. However this is only possible for $p_0$ or $p_0 + d$, as $p < 2d$. In this case, $p_0 + pd$ or $p_0 + (p+1)d$, respectively, is divisible by $p$ too and thus not prime.
Thus we have shown that the difference $d$ can occur at most $p+1$ times successively, and thus if the sequence of differences is not decreasing it can occur at most $p+1$ times in total. Recall that $p+1 le 2d$. So, the difference $d$ occurs at most $2d$ times.
This implies that the size of the $1+ sum_{d=1}^D 2d$-th prime is at least
$2+ sum_{d=1}^D (2d)d$.
Now the former sum is $1 + D(D+1)$ and the latter sum is $2 + frac{D(D+1)(2D+1)}{3}$. This later expression is at least of size $D^3/2$ (leaving very small $D$ aside).
Yet by the prime number theorem we know that there are about $D^3/2/log (D^3/2)$ primes below $D^3/2$. For sufficiently large $D$ this is much larger than $1 + D(D+1)$, and thus yields a contradiction to the $1 + D(D+1)$-th prime being greater than $D^3/2$.
$endgroup$
In brief: You can tweak the linked to argument. All you need in addition is a reasonable bound on the number of consecutive primes that have the same difference. It is not hard to show a bound linear in the difference.
Then proceed as in the linked question, that is show that this leads to an upper bound on the number of primes below a certain threshold that contradicts well-known results.
There it is that the number of primes below $N^2$ is at most linear in $N$. Here it will be that the number of primes below $N^2$ is at most quadratic in $N$. Both are absurd.
In detail:
Let us first show that the number of consecutive primes with difference $d$ is at most $2d$.
A difference $d$ is fixed. Let $p$ be a an auxiliary prime that does not divide $d$, for example the next prime after $d$ (which by Bertrand's postulate is $<2d$).
We show that there can be at most $p+2$ consecutive primes with difference $d$.
Let us consider $p_0, p_0 + d , p_0 + 2d , dots, p_0 + p d, p_0 + (p+1)d$. modulo $p$. We show at least one of these numbers is not prime.
Since $p nmid d$, we get that $p_0, p_0 + d , p_0 + 2d , dots,p_0 + (p-1)d$ covers all congruence classes modulo $p$. Thus in particular one of these is $0$ modulo $p$. This means it is not a prime unless it is equal to $p$ itself. However this is only possible for $p_0$ or $p_0 + d$, as $p < 2d$. In this case, $p_0 + pd$ or $p_0 + (p+1)d$, respectively, is divisible by $p$ too and thus not prime.
Thus we have shown that the difference $d$ can occur at most $p+1$ times successively, and thus if the sequence of differences is not decreasing it can occur at most $p+1$ times in total. Recall that $p+1 le 2d$. So, the difference $d$ occurs at most $2d$ times.
This implies that the size of the $1+ sum_{d=1}^D 2d$-th prime is at least
$2+ sum_{d=1}^D (2d)d$.
Now the former sum is $1 + D(D+1)$ and the latter sum is $2 + frac{D(D+1)(2D+1)}{3}$. This later expression is at least of size $D^3/2$ (leaving very small $D$ aside).
Yet by the prime number theorem we know that there are about $D^3/2/log (D^3/2)$ primes below $D^3/2$. For sufficiently large $D$ this is much larger than $1 + D(D+1)$, and thus yields a contradiction to the $1 + D(D+1)$-th prime being greater than $D^3/2$.
edited Feb 14 '16 at 22:02
answered Feb 10 '16 at 15:37
quid♦quid
37.1k95093
37.1k95093
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
1
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
|
show 7 more comments
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
1
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
I'm sorry but I don't see this, can anyone provide a reference please?
$endgroup$
– Brennan.Tobias
Feb 11 '16 at 22:56
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
What don't you see? Do you understand the argument linked?
$endgroup$
– quid♦
Feb 12 '16 at 0:09
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
I am a bit puzzled why you place a bounty rather than just answering my question. Anyway I'll write some details soon.
$endgroup$
– quid♦
Feb 14 '16 at 16:26
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
$begingroup$
There are more details now. Let me know what if anything is not clear.
$endgroup$
– quid♦
Feb 14 '16 at 22:06
1
1
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
$begingroup$
Sorry for the delay in responding. Yes the basic reasoning is as you say. On the specific numbers, I fumbled that a bit. But as there are $p+2$ primes there are $p+1$ differences between them, and this is already impossible, so in fact we could get down to $p$. But you are right it is better to argue this with $d+1$. Anything coprime to $d$ will do.
$endgroup$
– quid♦
Feb 15 '16 at 18:24
|
show 7 more comments
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$begingroup$
Presumably, you mean "not eventually monotonic." Because it can clearly be demonstrated to not be monotonic. Such as $13-11<11-7$.
$endgroup$
– Thomas Andrews
Feb 10 '16 at 15:26
$begingroup$
How about that $[p#, p#+p]$ are all composite. Then all you need is that some gap is less than $p$
$endgroup$
– user334732
Dec 24 '18 at 23:38