Prime numbers falling in the gap between twice the members of twin primes
up vote
1
down vote
favorite
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 at 20:06
Keith Backman
8501510
add a comment |
up vote
1
down vote
favorite
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 at 20:06
Keith Backman
8501510
"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)
– Hagen von Eitzen
Nov 23 at 20:16
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.'
– Keith Backman
Nov 23 at 20:27
$659$ and $809$ work
– TheSimpliFire
Nov 23 at 20:44
@TheSimpliFire Great find! Thanks.
– Keith Backman
Nov 23 at 20:45
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.
– TheSimpliFire
Nov 23 at 20:49
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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 at 20:06
Keith Backman
8501510
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 at 20:06
Keith Backman
8501510
asked Nov 23 at 20:06
Keith Backman
8501510
asked Nov 23 at 20:06
Keith Backman
8501510
asked Nov 23 at 20:06
Keith Backman
8501510
asked Nov 23 at 20:06
Keith Backman
8501510
8501510
"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)
– Hagen von Eitzen
Nov 23 at 20:16
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.'
– Keith Backman
Nov 23 at 20:27
$659$ and $809$ work
– TheSimpliFire
Nov 23 at 20:44
@TheSimpliFire Great find! Thanks.
– Keith Backman
Nov 23 at 20:45
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.
– TheSimpliFire
Nov 23 at 20:49
add a comment |
"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)
– Hagen von Eitzen
Nov 23 at 20:16
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.'
– Keith Backman
Nov 23 at 20:27
$659$ and $809$ work
– TheSimpliFire
Nov 23 at 20:44
@TheSimpliFire Great find! Thanks.
– Keith Backman
Nov 23 at 20:45
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.
– TheSimpliFire
Nov 23 at 20:49
"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)
– Hagen von Eitzen
Nov 23 at 20:16
"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)
– Hagen von Eitzen
Nov 23 at 20:16
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.'
– Keith Backman
Nov 23 at 20:27
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.'
– Keith Backman
Nov 23 at 20:27
$659$ and $809$ work
– TheSimpliFire
Nov 23 at 20:44
$659$ and $809$ work
– TheSimpliFire
Nov 23 at 20:44
@TheSimpliFire Great find! Thanks.
– Keith Backman
Nov 23 at 20:45
@TheSimpliFire Great find! Thanks.
– Keith Backman
Nov 23 at 20:45
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.
– TheSimpliFire
Nov 23 at 20:49
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.
– TheSimpliFire
Nov 23 at 20:49
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010776%2fprime-numbers-falling-in-the-gap-between-twice-the-members-of-twin-primes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010776%2fprime-numbers-falling-in-the-gap-between-twice-the-members-of-twin-primes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
"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)
– Hagen von Eitzen
Nov 23 at 20:16
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.'
– Keith Backman
Nov 23 at 20:27
$659$ and $809$ work
– TheSimpliFire
Nov 23 at 20:44
@TheSimpliFire Great find! Thanks.
– Keith Backman
Nov 23 at 20:45
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.
– TheSimpliFire
Nov 23 at 20:49