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







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$endgroup$












  • $begingroup$
    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











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1 Answer
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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


















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