Twin Primes Conjecture and related problemsGoldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primesTwin, cousin, sexy, … primesDisprove the Twin Prime Conjecture for Exotic PrimesSiamese Twin primesSmallest twin-prime-pair exceeding $10^1000$How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random?Proof of minor claim related to the Twin Primes ConjectureA twin prime theorem, and a reformulation of the twin prime conjecture
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Twin Primes Conjecture and related problems
Goldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primesTwin, cousin, sexy, … primesDisprove the Twin Prime Conjecture for Exotic PrimesSiamese Twin primesSmallest twin-prime-pair exceeding $10^1000$How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random?Proof of minor claim related to the Twin Primes ConjectureA twin prime theorem, and a reformulation of the twin prime conjecture
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty
margin-bottom:0;
.everyonelovesstackoverflowposition:absolute;height:1px;width:1px;opacity:0;top:0;left:0;pointer-events:none;
$begingroup$
I am wondering if my approach below, based on probability theory and thus heuristic (it is by no means a proof) is new and worth pursuing. Applications are numerous.
Let $S_f$ be a random set of positive integers defined as follows: the positive integer $n$ belongs to $S_f$ with probability $f(n)$. In other words, for each integer $n$, generate a uniform deviate $U_n$ on $[0, 1]$. If and only if $U_n < f(n)$ then add $n$ to $S_f$. The deviates $U_n$ are independent. Now consider $f(n) = frac1log n$ if $n > 2$, and $f(1)= f(2) = 0$. With this particular choice of $f$, the set $S_f$ has the same density as the set of prime numbers, and behaves in a very similar way. Let us define the following functions (actually, they are random variables):
$M(n)$ is the number of elements in $S_f$ that are smaller than $n$.
$T(n)$ is the number of twin pairs $(k, k+1)$ with both $k, k+1 in
S_f$ and $k leq n$.
The twins in $S_f$ play the same role as twin primes. It is is easy to prove that $M(n)$ (its expectation) is asymptotically equal to the logarithm integral $mboxLi(n)$. The same is true for the prime number counting function. Furthermore, the prime numbers themselves is a particular case of $S_f$, a particular realization of the random set $S_f$ for the same function $f$. (you need to ignore even integers in the construction process, but other than that, it's straightforward.)
Then we also have:
$$mboxE[T(n)] sim Csum_k=1^n-1 f(k)f(k+1)sim C'int_2^n fracdt(log t)^2 .$$
Here E denotes the expectation. The same result is conjectured to be true for twin primes. It is easy to show that there are infinitely many twin numbers in $S_f$, and assess whether the series consisting of the reciprocals of these twins, converge. The same arguments could be applied to twin primes, but of course, it would no constitute a proof, just some heuristic argumentation. Are there any useful references related to the approach discussed here?
Update: I plan on using the same methodology for the Goldbach conjecture, and in particular to obtain an asymptotic value for the expected number of ways an even number $n$ can be represented as a sum of two primes. With my notation, it should be asymptotically equivalent to
$$sum_k=1^lfloor n/2 rfloor f(k)f(n-k) sim fracn2(log n)^2.$$
See also here and here.
sequences-and-series number-theory probability-theory prime-numbers
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
$endgroup$
add a comment
|
$begingroup$
I am wondering if my approach below, based on probability theory and thus heuristic (it is by no means a proof) is new and worth pursuing. Applications are numerous.
Let $S_f$ be a random set of positive integers defined as follows: the positive integer $n$ belongs to $S_f$ with probability $f(n)$. In other words, for each integer $n$, generate a uniform deviate $U_n$ on $[0, 1]$. If and only if $U_n < f(n)$ then add $n$ to $S_f$. The deviates $U_n$ are independent. Now consider $f(n) = frac1log n$ if $n > 2$, and $f(1)= f(2) = 0$. With this particular choice of $f$, the set $S_f$ has the same density as the set of prime numbers, and behaves in a very similar way. Let us define the following functions (actually, they are random variables):
$M(n)$ is the number of elements in $S_f$ that are smaller than $n$.
$T(n)$ is the number of twin pairs $(k, k+1)$ with both $k, k+1 in
S_f$ and $k leq n$.
The twins in $S_f$ play the same role as twin primes. It is is easy to prove that $M(n)$ (its expectation) is asymptotically equal to the logarithm integral $mboxLi(n)$. The same is true for the prime number counting function. Furthermore, the prime numbers themselves is a particular case of $S_f$, a particular realization of the random set $S_f$ for the same function $f$. (you need to ignore even integers in the construction process, but other than that, it's straightforward.)
Then we also have:
$$mboxE[T(n)] sim Csum_k=1^n-1 f(k)f(k+1)sim C'int_2^n fracdt(log t)^2 .$$
Here E denotes the expectation. The same result is conjectured to be true for twin primes. It is easy to show that there are infinitely many twin numbers in $S_f$, and assess whether the series consisting of the reciprocals of these twins, converge. The same arguments could be applied to twin primes, but of course, it would no constitute a proof, just some heuristic argumentation. Are there any useful references related to the approach discussed here?
Update: I plan on using the same methodology for the Goldbach conjecture, and in particular to obtain an asymptotic value for the expected number of ways an even number $n$ can be represented as a sum of two primes. With my notation, it should be asymptotically equivalent to
$$sum_k=1^lfloor n/2 rfloor f(k)f(n-k) sim fracn2(log n)^2.$$
See also here and here.
sequences-and-series number-theory probability-theory prime-numbers
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
$endgroup$
$begingroup$
I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/…
$endgroup$
– LAGRIDA
Sep 29 at 13:09
add a comment
|
$begingroup$
I am wondering if my approach below, based on probability theory and thus heuristic (it is by no means a proof) is new and worth pursuing. Applications are numerous.
Let $S_f$ be a random set of positive integers defined as follows: the positive integer $n$ belongs to $S_f$ with probability $f(n)$. In other words, for each integer $n$, generate a uniform deviate $U_n$ on $[0, 1]$. If and only if $U_n < f(n)$ then add $n$ to $S_f$. The deviates $U_n$ are independent. Now consider $f(n) = frac1log n$ if $n > 2$, and $f(1)= f(2) = 0$. With this particular choice of $f$, the set $S_f$ has the same density as the set of prime numbers, and behaves in a very similar way. Let us define the following functions (actually, they are random variables):
$M(n)$ is the number of elements in $S_f$ that are smaller than $n$.
$T(n)$ is the number of twin pairs $(k, k+1)$ with both $k, k+1 in
S_f$ and $k leq n$.
The twins in $S_f$ play the same role as twin primes. It is is easy to prove that $M(n)$ (its expectation) is asymptotically equal to the logarithm integral $mboxLi(n)$. The same is true for the prime number counting function. Furthermore, the prime numbers themselves is a particular case of $S_f$, a particular realization of the random set $S_f$ for the same function $f$. (you need to ignore even integers in the construction process, but other than that, it's straightforward.)
Then we also have:
$$mboxE[T(n)] sim Csum_k=1^n-1 f(k)f(k+1)sim C'int_2^n fracdt(log t)^2 .$$
Here E denotes the expectation. The same result is conjectured to be true for twin primes. It is easy to show that there are infinitely many twin numbers in $S_f$, and assess whether the series consisting of the reciprocals of these twins, converge. The same arguments could be applied to twin primes, but of course, it would no constitute a proof, just some heuristic argumentation. Are there any useful references related to the approach discussed here?
Update: I plan on using the same methodology for the Goldbach conjecture, and in particular to obtain an asymptotic value for the expected number of ways an even number $n$ can be represented as a sum of two primes. With my notation, it should be asymptotically equivalent to
$$sum_k=1^lfloor n/2 rfloor f(k)f(n-k) sim fracn2(log n)^2.$$
See also here and here.
sequences-and-series number-theory probability-theory prime-numbers
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
$endgroup$
I am wondering if my approach below, based on probability theory and thus heuristic (it is by no means a proof) is new and worth pursuing. Applications are numerous.
Let $S_f$ be a random set of positive integers defined as follows: the positive integer $n$ belongs to $S_f$ with probability $f(n)$. In other words, for each integer $n$, generate a uniform deviate $U_n$ on $[0, 1]$. If and only if $U_n < f(n)$ then add $n$ to $S_f$. The deviates $U_n$ are independent. Now consider $f(n) = frac1log n$ if $n > 2$, and $f(1)= f(2) = 0$. With this particular choice of $f$, the set $S_f$ has the same density as the set of prime numbers, and behaves in a very similar way. Let us define the following functions (actually, they are random variables):
$M(n)$ is the number of elements in $S_f$ that are smaller than $n$.
$T(n)$ is the number of twin pairs $(k, k+1)$ with both $k, k+1 in
S_f$ and $k leq n$.
The twins in $S_f$ play the same role as twin primes. It is is easy to prove that $M(n)$ (its expectation) is asymptotically equal to the logarithm integral $mboxLi(n)$. The same is true for the prime number counting function. Furthermore, the prime numbers themselves is a particular case of $S_f$, a particular realization of the random set $S_f$ for the same function $f$. (you need to ignore even integers in the construction process, but other than that, it's straightforward.)
Then we also have:
$$mboxE[T(n)] sim Csum_k=1^n-1 f(k)f(k+1)sim C'int_2^n fracdt(log t)^2 .$$
Here E denotes the expectation. The same result is conjectured to be true for twin primes. It is easy to show that there are infinitely many twin numbers in $S_f$, and assess whether the series consisting of the reciprocals of these twins, converge. The same arguments could be applied to twin primes, but of course, it would no constitute a proof, just some heuristic argumentation. Are there any useful references related to the approach discussed here?
Update: I plan on using the same methodology for the Goldbach conjecture, and in particular to obtain an asymptotic value for the expected number of ways an even number $n$ can be represented as a sum of two primes. With my notation, it should be asymptotically equivalent to
$$sum_k=1^lfloor n/2 rfloor f(k)f(n-k) sim fracn2(log n)^2.$$
See also here and here.
sequences-and-series number-theory probability-theory prime-numbers
sequences-and-series number-theory probability-theory prime-numbers
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
edited Sep 29 at 12:42
Vincent Granville
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
asked Sep 29 at 0:42
Vincent GranvilleVincent Granville
2,0054 silver badges18 bronze badges
2,0054 silver badges18 bronze badges
$begingroup$
I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/…
$endgroup$
– LAGRIDA
Sep 29 at 13:09
add a comment
|
$begingroup$
I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/…
$endgroup$
– LAGRIDA
Sep 29 at 13:09
$begingroup$
I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/…
$endgroup$
– LAGRIDA
Sep 29 at 13:09
$begingroup$
I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/…
$endgroup$
– LAGRIDA
Sep 29 at 13:09
add a comment
|
1 Answer
1
active
oldest
votes
$begingroup$
You're about 90 years too late. The probabilistic approach to prime numbers was developed by Cramér in the 1930's. You might look at this paper by the other Granville.
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
$endgroup$
2
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
2
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
add a comment
|
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$begingroup$
You're about 90 years too late. The probabilistic approach to prime numbers was developed by Cramér in the 1930's. You might look at this paper by the other Granville.
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
$endgroup$
2
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
2
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
add a comment
|
$begingroup$
You're about 90 years too late. The probabilistic approach to prime numbers was developed by Cramér in the 1930's. You might look at this paper by the other Granville.
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
$endgroup$
2
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
2
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
add a comment
|
$begingroup$
You're about 90 years too late. The probabilistic approach to prime numbers was developed by Cramér in the 1930's. You might look at this paper by the other Granville.
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
$endgroup$
You're about 90 years too late. The probabilistic approach to prime numbers was developed by Cramér in the 1930's. You might look at this paper by the other Granville.
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
answered Sep 29 at 1:43
Robert IsraelRobert Israel
355k23 gold badges246 silver badges514 bronze badges
355k23 gold badges246 silver badges514 bronze badges
2
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
2
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
add a comment
|
2
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
2
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
2
2
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
$begingroup$
I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation.
$endgroup$
– Vincent Granville
Sep 29 at 2:01
2
2
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
$begingroup$
Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/…
$endgroup$
– Vincent Granville
Sep 29 at 2:10
add a comment
|
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$begingroup$
I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/…
$endgroup$
– LAGRIDA
Sep 29 at 13:09