Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the...












1












$begingroup$


Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$?



For example,
If we choose prime number 499, then $n=998$.
For $n = 998$ there are 33 prime pairs and there are 35 twin primes less than 998.



If we choose larger values of $n=2p$, the number of prime pairs will converge to the number of twin primes less than $n$?






number-theory prime-numbers prime-twins goldbachs-conjecture







share|cite|improve this question











$endgroup$












  • $begingroup$
    This may be a reasonable conjecture. However proving it will be quite difficult - just proving that the number of prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$ is always nonzero is Goldbach's conjecture, a long-standing unsolved problem.
    $endgroup$
    – Michael Lugo
    Dec 12 '18 at 19:26
















1












$begingroup$


Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$?



For example,
If we choose prime number 499, then $n=998$.
For $n = 998$ there are 33 prime pairs and there are 35 twin primes less than 998.



If we choose larger values of $n=2p$, the number of prime pairs will converge to the number of twin primes less than $n$?






number-theory prime-numbers prime-twins goldbachs-conjecture







share|cite|improve this question











$endgroup$












  • $begingroup$
    This may be a reasonable conjecture. However proving it will be quite difficult - just proving that the number of prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$ is always nonzero is Goldbach's conjecture, a long-standing unsolved problem.
    $endgroup$
    – Michael Lugo
    Dec 12 '18 at 19:26














1












1








1





$begingroup$


Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$?



For example,
If we choose prime number 499, then $n=998$.
For $n = 998$ there are 33 prime pairs and there are 35 twin primes less than 998.



If we choose larger values of $n=2p$, the number of prime pairs will converge to the number of twin primes less than $n$?






number-theory prime-numbers prime-twins goldbachs-conjecture







share|cite|improve this question











$endgroup$




Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$?



For example,
If we choose prime number 499, then $n=998$.
For $n = 998$ there are 33 prime pairs and there are 35 twin primes less than 998.



If we choose larger values of $n=2p$, the number of prime pairs will converge to the number of twin primes less than $n$?






number-theory prime-numbers prime-twins goldbachs-conjecture




number-theory prime-numbers prime-twins goldbachs-conjecture






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 19:20







temp watts

















asked Dec 12 '18 at 19:15









temp wattstemp watts

373




373












  • $begingroup$
    This may be a reasonable conjecture. However proving it will be quite difficult - just proving that the number of prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$ is always nonzero is Goldbach's conjecture, a long-standing unsolved problem.
    $endgroup$
    – Michael Lugo
    Dec 12 '18 at 19:26


















  • $begingroup$
    This may be a reasonable conjecture. However proving it will be quite difficult - just proving that the number of prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$ is always nonzero is Goldbach's conjecture, a long-standing unsolved problem.
    $endgroup$
    – Michael Lugo
    Dec 12 '18 at 19:26
















$begingroup$
This may be a reasonable conjecture. However proving it will be quite difficult - just proving that the number of prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$ is always nonzero is Goldbach's conjecture, a long-standing unsolved problem.
$endgroup$
– Michael Lugo
Dec 12 '18 at 19:26




$begingroup$
This may be a reasonable conjecture. However proving it will be quite difficult - just proving that the number of prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$ is always nonzero is Goldbach's conjecture, a long-standing unsolved problem.
$endgroup$
– Michael Lugo
Dec 12 '18 at 19:26










1 Answer
1






active

oldest

votes


















3












$begingroup$

The Hardy-Littlewood conjecture would tell us that the number of primes $p$ less than $x$ such that $p+2$ is also a prime should asymptotically satisfy: $$pi_2(x)sim 2C_2frac x{(ln x)^2}$$



Where $C_2approx .66$



On the other hand, standard heuristics tell us that the number of ways to express an even $x$ as the sum of two primes should be asymptotically $$2C_2times prod_{p,|,x,;p≥3}frac {p-1}{p-2}times frac x{(ln x)^2}$$



See, e.g., this.



Thus, standard conjectures would tell us that, for large even $x$:



$$frac {text {the number of prime pairs} ≤ x}{text {the number of ways to write} ;x;text{as the sum of two primes}}sim prod_{p,|,x,;p≥3}frac {p-2}{p-1}$$



To be sure, both conjectures are entirely unproven.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
    $endgroup$
    – temp watts
    Dec 13 '18 at 14:21










  • $begingroup$
    Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:04










  • $begingroup$
    They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:07











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3037104%2fis-it-true-for-any-n-2p-where-p-is-prime-that-the-number-of-twin-primes-les%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

The Hardy-Littlewood conjecture would tell us that the number of primes $p$ less than $x$ such that $p+2$ is also a prime should asymptotically satisfy: $$pi_2(x)sim 2C_2frac x{(ln x)^2}$$



Where $C_2approx .66$



On the other hand, standard heuristics tell us that the number of ways to express an even $x$ as the sum of two primes should be asymptotically $$2C_2times prod_{p,|,x,;p≥3}frac {p-1}{p-2}times frac x{(ln x)^2}$$



See, e.g., this.



Thus, standard conjectures would tell us that, for large even $x$:



$$frac {text {the number of prime pairs} ≤ x}{text {the number of ways to write} ;x;text{as the sum of two primes}}sim prod_{p,|,x,;p≥3}frac {p-2}{p-1}$$



To be sure, both conjectures are entirely unproven.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
    $endgroup$
    – temp watts
    Dec 13 '18 at 14:21










  • $begingroup$
    Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:04










  • $begingroup$
    They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:07
















3












$begingroup$

The Hardy-Littlewood conjecture would tell us that the number of primes $p$ less than $x$ such that $p+2$ is also a prime should asymptotically satisfy: $$pi_2(x)sim 2C_2frac x{(ln x)^2}$$



Where $C_2approx .66$



On the other hand, standard heuristics tell us that the number of ways to express an even $x$ as the sum of two primes should be asymptotically $$2C_2times prod_{p,|,x,;p≥3}frac {p-1}{p-2}times frac x{(ln x)^2}$$



See, e.g., this.



Thus, standard conjectures would tell us that, for large even $x$:



$$frac {text {the number of prime pairs} ≤ x}{text {the number of ways to write} ;x;text{as the sum of two primes}}sim prod_{p,|,x,;p≥3}frac {p-2}{p-1}$$



To be sure, both conjectures are entirely unproven.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
    $endgroup$
    – temp watts
    Dec 13 '18 at 14:21










  • $begingroup$
    Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:04










  • $begingroup$
    They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:07














3












3








3





$begingroup$

The Hardy-Littlewood conjecture would tell us that the number of primes $p$ less than $x$ such that $p+2$ is also a prime should asymptotically satisfy: $$pi_2(x)sim 2C_2frac x{(ln x)^2}$$



Where $C_2approx .66$



On the other hand, standard heuristics tell us that the number of ways to express an even $x$ as the sum of two primes should be asymptotically $$2C_2times prod_{p,|,x,;p≥3}frac {p-1}{p-2}times frac x{(ln x)^2}$$



See, e.g., this.



Thus, standard conjectures would tell us that, for large even $x$:



$$frac {text {the number of prime pairs} ≤ x}{text {the number of ways to write} ;x;text{as the sum of two primes}}sim prod_{p,|,x,;p≥3}frac {p-2}{p-1}$$



To be sure, both conjectures are entirely unproven.






share|cite|improve this answer











$endgroup$



The Hardy-Littlewood conjecture would tell us that the number of primes $p$ less than $x$ such that $p+2$ is also a prime should asymptotically satisfy: $$pi_2(x)sim 2C_2frac x{(ln x)^2}$$



Where $C_2approx .66$



On the other hand, standard heuristics tell us that the number of ways to express an even $x$ as the sum of two primes should be asymptotically $$2C_2times prod_{p,|,x,;p≥3}frac {p-1}{p-2}times frac x{(ln x)^2}$$



See, e.g., this.



Thus, standard conjectures would tell us that, for large even $x$:



$$frac {text {the number of prime pairs} ≤ x}{text {the number of ways to write} ;x;text{as the sum of two primes}}sim prod_{p,|,x,;p≥3}frac {p-2}{p-1}$$



To be sure, both conjectures are entirely unproven.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 12 '18 at 20:52

























answered Dec 12 '18 at 20:46









lulululu

40.9k24879




40.9k24879












  • $begingroup$
    Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
    $endgroup$
    – temp watts
    Dec 13 '18 at 14:21










  • $begingroup$
    Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:04










  • $begingroup$
    They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:07


















  • $begingroup$
    Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
    $endgroup$
    – temp watts
    Dec 13 '18 at 14:21










  • $begingroup$
    Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:04










  • $begingroup$
    They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
    $endgroup$
    – lulu
    Dec 13 '18 at 15:07
















$begingroup$
Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
$endgroup$
– temp watts
Dec 13 '18 at 14:21




$begingroup$
Thank you for your response. I am only looking for values of x that are equal to 2p where p is a prime number.
$endgroup$
– temp watts
Dec 13 '18 at 14:21












$begingroup$
Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
$endgroup$
– lulu
Dec 13 '18 at 15:04




$begingroup$
Well, my response covers that case. If $x=2p$ then that product is just $frac {p-2}{p-1}$.
$endgroup$
– lulu
Dec 13 '18 at 15:04












$begingroup$
They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
$endgroup$
– lulu
Dec 13 '18 at 15:07




$begingroup$
They key point, of course, is that neither of these conjectures is even close to being proven. There are sensible heuristic arguments for both, but so what? Neither Twin Primes nor Goldbach is at all well understood so the heuristic arguments might entirely miss the point.
$endgroup$
– lulu
Dec 13 '18 at 15:07


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3037104%2fis-it-true-for-any-n-2p-where-p-is-prime-that-the-number-of-twin-primes-les%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

Quarter-circle Tiles

Image
up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
Read more

build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

Image
0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
Read more

Mont Emei

Image
Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
Read more