Estimation on Primorial Influence
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2
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The primorial ($#$) function is defined as the product of first $n$ prime numbers. That is,
$$
n# = prod_{i=1}^nP_i
$$
And there is some effect named Primorial Influence (explained here)which prevents the numbers near to a Primorial to be prime. That is,except than $n# + 1$ or $n# - 1$ which are Primorial prime for sum $n$, $n# + c$ can be prime only if $c ge P_{n+1}$. As an example the below near Primorial numbers are prime:
$$
P_{1000}# + P_{1087} implies text{3393 digits, }P_{1087} = 8719 \
P_{1001}# + P_{1100} implies text{3397 digits, }P_{1100} = 8831 \
P_{1002}# + P_{1068} implies text{3401 digits, }P_{1068} = 8573
$$
From samples above it can be seen that difference of prime index of $n$ and $c$ is less than $100$. I know it can't be generalized to any primorial, but isn't there any good estimation on this upper bound?
number-theory prime-numbers
edited Nov 21 at 12:00
Flermat
1,28311129
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
add a comment |
up vote
2
down vote
favorite
The primorial ($#$) function is defined as the product of first $n$ prime numbers. That is,
$$
n# = prod_{i=1}^nP_i
$$
And there is some effect named Primorial Influence (explained here)which prevents the numbers near to a Primorial to be prime. That is,except than $n# + 1$ or $n# - 1$ which are Primorial prime for sum $n$, $n# + c$ can be prime only if $c ge P_{n+1}$. As an example the below near Primorial numbers are prime:
$$
P_{1000}# + P_{1087} implies text{3393 digits, }P_{1087} = 8719 \
P_{1001}# + P_{1100} implies text{3397 digits, }P_{1100} = 8831 \
P_{1002}# + P_{1068} implies text{3401 digits, }P_{1068} = 8573
$$
From samples above it can be seen that difference of prime index of $n$ and $c$ is less than $100$. I know it can't be generalized to any primorial, but isn't there any good estimation on this upper bound?
number-theory prime-numbers
edited Nov 21 at 12:00
Flermat
1,28311129
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
$2# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example?
– user51427
Dec 9 '12 at 22:33
1
@sunflower, you're true, I forgot to add $n# + 1$ and $n# - 1$ special cases. Question updated.
– Mohsen Afshin
Dec 10 '12 at 6:19
1
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+sqrt n$.
– Gerry Myerson
Dec 10 '12 at 6:30
The prime number theorem implies $n#approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets.
– Jaycob Coleman
Nov 8 '13 at 3:43
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The primorial ($#$) function is defined as the product of first $n$ prime numbers. That is,
$$
n# = prod_{i=1}^nP_i
$$
And there is some effect named Primorial Influence (explained here)which prevents the numbers near to a Primorial to be prime. That is,except than $n# + 1$ or $n# - 1$ which are Primorial prime for sum $n$, $n# + c$ can be prime only if $c ge P_{n+1}$. As an example the below near Primorial numbers are prime:
$$
P_{1000}# + P_{1087} implies text{3393 digits, }P_{1087} = 8719 \
P_{1001}# + P_{1100} implies text{3397 digits, }P_{1100} = 8831 \
P_{1002}# + P_{1068} implies text{3401 digits, }P_{1068} = 8573
$$
From samples above it can be seen that difference of prime index of $n$ and $c$ is less than $100$. I know it can't be generalized to any primorial, but isn't there any good estimation on this upper bound?
number-theory prime-numbers
edited Nov 21 at 12:00
Flermat
1,28311129
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
The primorial ($#$) function is defined as the product of first $n$ prime numbers. That is,
$$
n# = prod_{i=1}^nP_i
$$
And there is some effect named Primorial Influence (explained here)which prevents the numbers near to a Primorial to be prime. That is,except than $n# + 1$ or $n# - 1$ which are Primorial prime for sum $n$, $n# + c$ can be prime only if $c ge P_{n+1}$. As an example the below near Primorial numbers are prime:
$$
P_{1000}# + P_{1087} implies text{3393 digits, }P_{1087} = 8719 \
P_{1001}# + P_{1100} implies text{3397 digits, }P_{1100} = 8831 \
P_{1002}# + P_{1068} implies text{3401 digits, }P_{1068} = 8573
$$
From samples above it can be seen that difference of prime index of $n$ and $c$ is less than $100$. I know it can't be generalized to any primorial, but isn't there any good estimation on this upper bound?
number-theory prime-numbers
number-theory prime-numbers
edited Nov 21 at 12:00
Flermat
1,28311129
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
edited Nov 21 at 12:00
Flermat
1,28311129
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
edited Nov 21 at 12:00
Flermat
1,28311129
edited Nov 21 at 12:00
Flermat
1,28311129
edited Nov 21 at 12:00
Flermat
1,28311129
1,28311129
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
asked Dec 9 '12 at 21:12
Mohsen Afshin
313314
313314
$2# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example?
– user51427
Dec 9 '12 at 22:33
1
@sunflower, you're true, I forgot to add $n# + 1$ and $n# - 1$ special cases. Question updated.
– Mohsen Afshin
Dec 10 '12 at 6:19
1
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+sqrt n$.
– Gerry Myerson
Dec 10 '12 at 6:30
The prime number theorem implies $n#approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets.
– Jaycob Coleman
Nov 8 '13 at 3:43
add a comment |
$2# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example?
– user51427
Dec 9 '12 at 22:33
1
@sunflower, you're true, I forgot to add $n# + 1$ and $n# - 1$ special cases. Question updated.
– Mohsen Afshin
Dec 10 '12 at 6:19
1
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+sqrt n$.
– Gerry Myerson
Dec 10 '12 at 6:30
The prime number theorem implies $n#approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets.
– Jaycob Coleman
Nov 8 '13 at 3:43
$2# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example?
– user51427
Dec 9 '12 at 22:33
$2# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example?
– user51427
Dec 9 '12 at 22:33
1
1
@sunflower, you're true, I forgot to add $n# + 1$ and $n# - 1$ special cases. Question updated.
– Mohsen Afshin
Dec 10 '12 at 6:19
@sunflower, you're true, I forgot to add $n# + 1$ and $n# - 1$ special cases. Question updated.
– Mohsen Afshin
Dec 10 '12 at 6:19
1
1
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+sqrt n$.
– Gerry Myerson
Dec 10 '12 at 6:30
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+sqrt n$.
– Gerry Myerson
Dec 10 '12 at 6:30
The prime number theorem implies $n#approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets.
– Jaycob Coleman
Nov 8 '13 at 3:43
The prime number theorem implies $n#approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets.
– Jaycob Coleman
Nov 8 '13 at 3:43
add a comment |
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$2# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example?
– user51427
Dec 9 '12 at 22:33
1
@sunflower, you're true, I forgot to add $n# + 1$ and $n# - 1$ special cases. Question updated.
– Mohsen Afshin
Dec 10 '12 at 6:19
1
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+sqrt n$.
– Gerry Myerson
Dec 10 '12 at 6:30
The prime number theorem implies $n#approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets.
– Jaycob Coleman
Nov 8 '13 at 3:43