Prime numbers falling in the gap between twice the members of twin primes
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Frequently, the gap between twice a pair of twin primes contains a prime number. That is, for $p_i,(p_i+2)in mathbb P$, it is often but not always the case that one of $2p_i+1$ or $2p_i+3$ is also prime. In fact, for $p_i=29$ both $59,61$ (another pair of twin primes) are prime.
I looked at the first sixty pairs of twin primes (by hand; I'm not a programmer). What I observed is that the number of instances in which a gap between twice a pair of twin primes fails to feature a prime tends to increase as the size of the twin primes increases, but the effect is not particularly rapid. $p_i$ for which $(p_i+2)in mathbb P$ and neither $2p_i+1$ nor $2p_i+3$ are also prime include: $71,103,107,109,149,311,347,461,521,569,821,857,881,1061,1091,1151,1301,1319,1487,1619,1667,1697,1721,1787,1871,1877,1949$
Minor question: Are there other instances where $p_i,(p_i+2)in mathbb P$, and both $2p_i+1$ and $2p_i+3$ are also prime?
Refinements to Bertrand's Postulate suggest that for arbitrarily small $epsilon$, there is always a value $n_0$ such that for $n>n_0$ there is a prime $p$ in the gap $n<p<(1+epsilon)n$. Various formulations of $epsilon$ with regard to various values of $n_0$ have been advanced. In the case I am looking at, $epsilon = frac{2}{p_i}$ which gets arbitrarily small as $p_i$ gets large. Depending on how rapidly the value of $epsilon$ in my scenario diminishes compared to other evaluations of $epsilon$, it might become the case that the gaps I am discussing either must always or may never contain a prime number. With regard to the 'must always' option, the data at small values of $p_i$ run counter to it in that primeless gaps occur and appear to increase in frequency as $p_i$ increases.
Main question: Is there a number $n_0$, such that $p_i>n_0$, $(p_i+2)in mathbb P$, and one of $2p_i+1$ or $2p_i+3$ must be prime?
number-theory prime-numbers prime-twins
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
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add a comment |
$begingroup$
Frequently, the gap between twice a pair of twin primes contains a prime number. That is, for $p_i,(p_i+2)in mathbb P$, it is often but not always the case that one of $2p_i+1$ or $2p_i+3$ is also prime. In fact, for $p_i=29$ both $59,61$ (another pair of twin primes) are prime.
I looked at the first sixty pairs of twin primes (by hand; I'm not a programmer). What I observed is that the number of instances in which a gap between twice a pair of twin primes fails to feature a prime tends to increase as the size of the twin primes increases, but the effect is not particularly rapid. $p_i$ for which $(p_i+2)in mathbb P$ and neither $2p_i+1$ nor $2p_i+3$ are also prime include: $71,103,107,109,149,311,347,461,521,569,821,857,881,1061,1091,1151,1301,1319,1487,1619,1667,1697,1721,1787,1871,1877,1949$
Minor question: Are there other instances where $p_i,(p_i+2)in mathbb P$, and both $2p_i+1$ and $2p_i+3$ are also prime?
Refinements to Bertrand's Postulate suggest that for arbitrarily small $epsilon$, there is always a value $n_0$ such that for $n>n_0$ there is a prime $p$ in the gap $n<p<(1+epsilon)n$. Various formulations of $epsilon$ with regard to various values of $n_0$ have been advanced. In the case I am looking at, $epsilon = frac{2}{p_i}$ which gets arbitrarily small as $p_i$ gets large. Depending on how rapidly the value of $epsilon$ in my scenario diminishes compared to other evaluations of $epsilon$, it might become the case that the gaps I am discussing either must always or may never contain a prime number. With regard to the 'must always' option, the data at small values of $p_i$ run counter to it in that primeless gaps occur and appear to increase in frequency as $p_i$ increases.
Main question: Is there a number $n_0$, such that $p_i>n_0$, $(p_i+2)in mathbb P$, and one of $2p_i+1$ or $2p_i+3$ must be prime?
number-theory prime-numbers prime-twins
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
$endgroup$
$begingroup$
"how rapidly the value of $epsilon$ ... diminishes" - In the generalized Bertrand postulate, $epsilon$ does not diminish at all (in the original Bertrand postulate, $epsilon=1$ does not diminish either)
$endgroup$
– Hagen von Eitzen
Nov 23 '18 at 20:16
$begingroup$
In the Wikipedia article en.wikipedia.org/wiki/Bertrand%27s_postulate in the section 'Better results' it gives successively lower values for $epsilon$ as $n_0$ increases. Perhaps my wording was inelegant, but my understanding is that no matter how small one chooses to set $epsilon$, one can find a range $n>n_0$ where $n<p<(1+epsilon)n$. I interpret that to imply that if one goes to larger $n$, one is free to choose smaller $epsilon$. That is what I meant by 'diminish.'
$endgroup$
– Keith Backman
Nov 23 '18 at 20:27
$begingroup$
$659$ and $809$ work
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:44
$begingroup$
@TheSimpliFire Great find! Thanks.
$endgroup$
– Keith Backman
Nov 23 '18 at 20:45
$begingroup$
I've added another one. But all I'm using is a table of primes with integers down the page and $1,3,7,9$ across which could be accessible on the web if you wish to find any more.
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:49
add a comment |
$begingroup$
Frequently, the gap between twice a pair of twin primes contains a prime number. That is, for $p_i,(p_i+2)in mathbb P$, it is often but not always the case that one of $2p_i+1$ or $2p_i+3$ is also prime. In fact, for $p_i=29$ both $59,61$ (another pair of twin primes) are prime.
I looked at the first sixty pairs of twin primes (by hand; I'm not a programmer). What I observed is that the number of instances in which a gap between twice a pair of twin primes fails to feature a prime tends to increase as the size of the twin primes increases, but the effect is not particularly rapid. $p_i$ for which $(p_i+2)in mathbb P$ and neither $2p_i+1$ nor $2p_i+3$ are also prime include: $71,103,107,109,149,311,347,461,521,569,821,857,881,1061,1091,1151,1301,1319,1487,1619,1667,1697,1721,1787,1871,1877,1949$
Minor question: Are there other instances where $p_i,(p_i+2)in mathbb P$, and both $2p_i+1$ and $2p_i+3$ are also prime?
Refinements to Bertrand's Postulate suggest that for arbitrarily small $epsilon$, there is always a value $n_0$ such that for $n>n_0$ there is a prime $p$ in the gap $n<p<(1+epsilon)n$. Various formulations of $epsilon$ with regard to various values of $n_0$ have been advanced. In the case I am looking at, $epsilon = frac{2}{p_i}$ which gets arbitrarily small as $p_i$ gets large. Depending on how rapidly the value of $epsilon$ in my scenario diminishes compared to other evaluations of $epsilon$, it might become the case that the gaps I am discussing either must always or may never contain a prime number. With regard to the 'must always' option, the data at small values of $p_i$ run counter to it in that primeless gaps occur and appear to increase in frequency as $p_i$ increases.
Main question: Is there a number $n_0$, such that $p_i>n_0$, $(p_i+2)in mathbb P$, and one of $2p_i+1$ or $2p_i+3$ must be prime?
number-theory prime-numbers prime-twins
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
$endgroup$
Frequently, the gap between twice a pair of twin primes contains a prime number. That is, for $p_i,(p_i+2)in mathbb P$, it is often but not always the case that one of $2p_i+1$ or $2p_i+3$ is also prime. In fact, for $p_i=29$ both $59,61$ (another pair of twin primes) are prime.
I looked at the first sixty pairs of twin primes (by hand; I'm not a programmer). What I observed is that the number of instances in which a gap between twice a pair of twin primes fails to feature a prime tends to increase as the size of the twin primes increases, but the effect is not particularly rapid. $p_i$ for which $(p_i+2)in mathbb P$ and neither $2p_i+1$ nor $2p_i+3$ are also prime include: $71,103,107,109,149,311,347,461,521,569,821,857,881,1061,1091,1151,1301,1319,1487,1619,1667,1697,1721,1787,1871,1877,1949$
Minor question: Are there other instances where $p_i,(p_i+2)in mathbb P$, and both $2p_i+1$ and $2p_i+3$ are also prime?
Refinements to Bertrand's Postulate suggest that for arbitrarily small $epsilon$, there is always a value $n_0$ such that for $n>n_0$ there is a prime $p$ in the gap $n<p<(1+epsilon)n$. Various formulations of $epsilon$ with regard to various values of $n_0$ have been advanced. In the case I am looking at, $epsilon = frac{2}{p_i}$ which gets arbitrarily small as $p_i$ gets large. Depending on how rapidly the value of $epsilon$ in my scenario diminishes compared to other evaluations of $epsilon$, it might become the case that the gaps I am discussing either must always or may never contain a prime number. With regard to the 'must always' option, the data at small values of $p_i$ run counter to it in that primeless gaps occur and appear to increase in frequency as $p_i$ increases.
Main question: Is there a number $n_0$, such that $p_i>n_0$, $(p_i+2)in mathbb P$, and one of $2p_i+1$ or $2p_i+3$ must be prime?
number-theory prime-numbers prime-twins
number-theory prime-numbers prime-twins
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
asked Nov 23 '18 at 20:06
Keith BackmanKeith Backman
1,1041712
1,1041712
$begingroup$
"how rapidly the value of $epsilon$ ... diminishes" - In the generalized Bertrand postulate, $epsilon$ does not diminish at all (in the original Bertrand postulate, $epsilon=1$ does not diminish either)
$endgroup$
– Hagen von Eitzen
Nov 23 '18 at 20:16
$begingroup$
In the Wikipedia article en.wikipedia.org/wiki/Bertrand%27s_postulate in the section 'Better results' it gives successively lower values for $epsilon$ as $n_0$ increases. Perhaps my wording was inelegant, but my understanding is that no matter how small one chooses to set $epsilon$, one can find a range $n>n_0$ where $n<p<(1+epsilon)n$. I interpret that to imply that if one goes to larger $n$, one is free to choose smaller $epsilon$. That is what I meant by 'diminish.'
$endgroup$
– Keith Backman
Nov 23 '18 at 20:27
$begingroup$
$659$ and $809$ work
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:44
$begingroup$
@TheSimpliFire Great find! Thanks.
$endgroup$
– Keith Backman
Nov 23 '18 at 20:45
$begingroup$
I've added another one. But all I'm using is a table of primes with integers down the page and $1,3,7,9$ across which could be accessible on the web if you wish to find any more.
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:49
add a comment |
$begingroup$
"how rapidly the value of $epsilon$ ... diminishes" - In the generalized Bertrand postulate, $epsilon$ does not diminish at all (in the original Bertrand postulate, $epsilon=1$ does not diminish either)
$endgroup$
– Hagen von Eitzen
Nov 23 '18 at 20:16
$begingroup$
In the Wikipedia article en.wikipedia.org/wiki/Bertrand%27s_postulate in the section 'Better results' it gives successively lower values for $epsilon$ as $n_0$ increases. Perhaps my wording was inelegant, but my understanding is that no matter how small one chooses to set $epsilon$, one can find a range $n>n_0$ where $n<p<(1+epsilon)n$. I interpret that to imply that if one goes to larger $n$, one is free to choose smaller $epsilon$. That is what I meant by 'diminish.'
$endgroup$
– Keith Backman
Nov 23 '18 at 20:27
$begingroup$
$659$ and $809$ work
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:44
$begingroup$
@TheSimpliFire Great find! Thanks.
$endgroup$
– Keith Backman
Nov 23 '18 at 20:45
$begingroup$
I've added another one. But all I'm using is a table of primes with integers down the page and $1,3,7,9$ across which could be accessible on the web if you wish to find any more.
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:49
$begingroup$
"how rapidly the value of $epsilon$ ... diminishes" - In the generalized Bertrand postulate, $epsilon$ does not diminish at all (in the original Bertrand postulate, $epsilon=1$ does not diminish either)
$endgroup$
– Hagen von Eitzen
Nov 23 '18 at 20:16
$begingroup$
"how rapidly the value of $epsilon$ ... diminishes" - In the generalized Bertrand postulate, $epsilon$ does not diminish at all (in the original Bertrand postulate, $epsilon=1$ does not diminish either)
$endgroup$
– Hagen von Eitzen
Nov 23 '18 at 20:16
$begingroup$
In the Wikipedia article en.wikipedia.org/wiki/Bertrand%27s_postulate in the section 'Better results' it gives successively lower values for $epsilon$ as $n_0$ increases. Perhaps my wording was inelegant, but my understanding is that no matter how small one chooses to set $epsilon$, one can find a range $n>n_0$ where $n<p<(1+epsilon)n$. I interpret that to imply that if one goes to larger $n$, one is free to choose smaller $epsilon$. That is what I meant by 'diminish.'
$endgroup$
– Keith Backman
Nov 23 '18 at 20:27
$begingroup$
In the Wikipedia article en.wikipedia.org/wiki/Bertrand%27s_postulate in the section 'Better results' it gives successively lower values for $epsilon$ as $n_0$ increases. Perhaps my wording was inelegant, but my understanding is that no matter how small one chooses to set $epsilon$, one can find a range $n>n_0$ where $n<p<(1+epsilon)n$. I interpret that to imply that if one goes to larger $n$, one is free to choose smaller $epsilon$. That is what I meant by 'diminish.'
$endgroup$
– Keith Backman
Nov 23 '18 at 20:27
$begingroup$
$659$ and $809$ work
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:44
$begingroup$
$659$ and $809$ work
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:44
$begingroup$
@TheSimpliFire Great find! Thanks.
$endgroup$
– Keith Backman
Nov 23 '18 at 20:45
$begingroup$
@TheSimpliFire Great find! Thanks.
$endgroup$
– Keith Backman
Nov 23 '18 at 20:45
$begingroup$
I've added another one. But all I'm using is a table of primes with integers down the page and $1,3,7,9$ across which could be accessible on the web if you wish to find any more.
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:49
$begingroup$
I've added another one. But all I'm using is a table of primes with integers down the page and $1,3,7,9$ across which could be accessible on the web if you wish to find any more.
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:49
add a comment |
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$begingroup$
"how rapidly the value of $epsilon$ ... diminishes" - In the generalized Bertrand postulate, $epsilon$ does not diminish at all (in the original Bertrand postulate, $epsilon=1$ does not diminish either)
$endgroup$
– Hagen von Eitzen
Nov 23 '18 at 20:16
$begingroup$
In the Wikipedia article en.wikipedia.org/wiki/Bertrand%27s_postulate in the section 'Better results' it gives successively lower values for $epsilon$ as $n_0$ increases. Perhaps my wording was inelegant, but my understanding is that no matter how small one chooses to set $epsilon$, one can find a range $n>n_0$ where $n<p<(1+epsilon)n$. I interpret that to imply that if one goes to larger $n$, one is free to choose smaller $epsilon$. That is what I meant by 'diminish.'
$endgroup$
– Keith Backman
Nov 23 '18 at 20:27
$begingroup$
$659$ and $809$ work
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:44
$begingroup$
@TheSimpliFire Great find! Thanks.
$endgroup$
– Keith Backman
Nov 23 '18 at 20:45
$begingroup$
I've added another one. But all I'm using is a table of primes with integers down the page and $1,3,7,9$ across which could be accessible on the web if you wish to find any more.
$endgroup$
– TheSimpliFire
Nov 23 '18 at 20:49