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Probability of Prime within radius around number

Which other unsolved problems, have necessary restrictions on the prime gaps?Are there infinitely many primes next to smooth numbers?Location of Prime Gaps SubsequenceIf $n-1$ is $f(n)$-smooth, $n$ is prime infinitely often. What is the best $f$?Is there a relationship between local prime gaps and cyclical graphs?Asymptotic density of Zhang's primesLongest sequence of primes where each term is obtained by appending a new digit to the previous termOn the Cramér-Granville Conjecture and finding prime pairs whose difference is 666On prime conjectures and the abc conjectureAre there an infinite number of primes of the form $lfloor pi n rfloor$?A conjecture regarding prime numbers

1

$begingroup$

Here is my question: do we have any kind of estimate about $p_k, d(n)$ the probability that there are at least $k$ prime numbers in a radius of $d$ around $n$?

Do you have any suggestions regarding related work?

For instance, we know that for $n$ there is a prime $p : nleq p leq 2n $ (Tchebychev, 1850), meaning:

$p_1, n/2(frac3n2) = 1, forall n>1$

Also since it has been shown that there are infinitely many prime gaps at most 246:

$p_1, 246(n) neq 0, forall n>1$

---

$^1$ I believe 246 is the smallest, though 2 is a well known conjecture

prime-numbers

share|cite|improve this question

edited Mar 12 at 21:32

ted

asked Mar 12 at 20:47

tedted

1084

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Chebyshev means rather that $p_1,n/2(3n/2)=1$. What you have written means that there is a prime between $0$ and $2n$.
  $endgroup$
  – TonyK
  Mar 12 at 20:51
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't think this is true, it says: there is at least one prime between n and 2n which means there is at least 1 prime within a radius n from n does it not?
  $endgroup$
  – ted
  Mar 12 at 20:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I am not sure that $246$ is known in the sense that you have put - it is known that there are an infinite number of prime gaps less than or equal to this - but it would be possible for there to be a finite number of gaps of size $246$ provided there were a smaller number (say $214$) for which the number of gaps of this size is infinite. At least on my reading of what is known.
  $endgroup$
  – Mark Bennet
  Mar 12 at 21:02
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  TonyK is saying that there is a prime within $n/2$ of $3n/2$ which halves the radius.
  $endgroup$
  – Mark Bennet
  Mar 12 at 21:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  There are arbitrarily large prime gaps, so there are some $n$ for which there are no primes in the interval $n-246, n+246$. For those $n$ your statement $p_1,246(n) ne 0$ is false. For example, $n = 500! + 248$.
  $endgroup$
  – Robert Israel
  Mar 12 at 21:38
  

  
  
  
  

 | 
show 9 more comments

1

$begingroup$

Here is my question: do we have any kind of estimate about $p_k, d(n)$ the probability that there are at least $k$ prime numbers in a radius of $d$ around $n$?

Do you have any suggestions regarding related work?

For instance, we know that for $n$ there is a prime $p : nleq p leq 2n $ (Tchebychev, 1850), meaning:

$p_1, n/2(frac3n2) = 1, forall n>1$

Also since it has been shown that there are infinitely many prime gaps at most 246:

$p_1, 246(n) neq 0, forall n>1$

---

$^1$ I believe 246 is the smallest, though 2 is a well known conjecture

prime-numbers

share|cite|improve this question

edited Mar 12 at 21:32

ted

asked Mar 12 at 20:47

tedted

1084

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Chebyshev means rather that $p_1,n/2(3n/2)=1$. What you have written means that there is a prime between $0$ and $2n$.
  $endgroup$
  – TonyK
  Mar 12 at 20:51
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't think this is true, it says: there is at least one prime between n and 2n which means there is at least 1 prime within a radius n from n does it not?
  $endgroup$
  – ted
  Mar 12 at 20:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I am not sure that $246$ is known in the sense that you have put - it is known that there are an infinite number of prime gaps less than or equal to this - but it would be possible for there to be a finite number of gaps of size $246$ provided there were a smaller number (say $214$) for which the number of gaps of this size is infinite. At least on my reading of what is known.
  $endgroup$
  – Mark Bennet
  Mar 12 at 21:02
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  TonyK is saying that there is a prime within $n/2$ of $3n/2$ which halves the radius.
  $endgroup$
  – Mark Bennet
  Mar 12 at 21:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  There are arbitrarily large prime gaps, so there are some $n$ for which there are no primes in the interval $n-246, n+246$. For those $n$ your statement $p_1,246(n) ne 0$ is false. For example, $n = 500! + 248$.
  $endgroup$
  – Robert Israel
  Mar 12 at 21:38
  

  
  
  
  

 | 
show 9 more comments

$begingroup$

Here is my question: do we have any kind of estimate about $p_k, d(n)$ the probability that there are at least $k$ prime numbers in a radius of $d$ around $n$?

Do you have any suggestions regarding related work?

For instance, we know that for $n$ there is a prime $p : nleq p leq 2n $ (Tchebychev, 1850), meaning:

$p_1, n/2(frac3n2) = 1, forall n>1$

Also since it has been shown that there are infinitely many prime gaps at most 246:

$p_1, 246(n) neq 0, forall n>1$

---

$^1$ I believe 246 is the smallest, though 2 is a well known conjecture

prime-numbers

share|cite|improve this question

edited Mar 12 at 21:32

ted

asked Mar 12 at 20:47

tedted

1084

$endgroup$

share|cite|improve this question

edited Mar 12 at 21:32

ted

asked Mar 12 at 20:47

tedted

1084

share|cite|improve this question

edited Mar 12 at 21:32

ted

asked Mar 12 at 20:47

tedted

1084

asked Mar 12 at 20:47

tedted

1084

asked Mar 12 at 20:47

tedted

1084

1 Answer
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oldest

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0

$begingroup$

Which other unsolved problems, have necessary restrictions on the prime gaps? a related question I just got answered. As the comments on your question talk about though, there's not really a restriction. Primorials (products of all primes up to a number) have potentially massive gaps nearby, You can gaurantee all numbers from the primorial plus or minus 2, until the primorial plus or minus the first prime not in the primorial minus or plus 1, are composite for 30=2*3*5 you get that all numbers in ranges 24-28 and 32-36 are necessarily composite (divisible by a prime in the factorization of 30). Unsolved conjectures, put bounds on d for all k values. Goldbach, has Bertrand's postulate as a necessary condition. Legendre, implies that two primes exists between $n^2$ and $(n+2)^2$ , n, a natural number. Grimm's, implies that $d<pi(n)$ for k=1, for almost all ( all but finitely many) n. If not then we have a pigeonhole contradiction.

share|cite|improve this answer

edited Mar 14 at 14:20

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116

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add a comment | 

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0

$begingroup$

Which other unsolved problems, have necessary restrictions on the prime gaps? a related question I just got answered. As the comments on your question talk about though, there's not really a restriction. Primorials (products of all primes up to a number) have potentially massive gaps nearby, You can gaurantee all numbers from the primorial plus or minus 2, until the primorial plus or minus the first prime not in the primorial minus or plus 1, are composite for 30=2*3*5 you get that all numbers in ranges 24-28 and 32-36 are necessarily composite (divisible by a prime in the factorization of 30). Unsolved conjectures, put bounds on d for all k values. Goldbach, has Bertrand's postulate as a necessary condition. Legendre, implies that two primes exists between $n^2$ and $(n+2)^2$ , n, a natural number. Grimm's, implies that $d<pi(n)$ for k=1, for almost all ( all but finitely many) n. If not then we have a pigeonhole contradiction.

share|cite|improve this answer

edited Mar 14 at 14:20

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116

$endgroup$

add a comment | 

0

$begingroup$

Which other unsolved problems, have necessary restrictions on the prime gaps? a related question I just got answered. As the comments on your question talk about though, there's not really a restriction. Primorials (products of all primes up to a number) have potentially massive gaps nearby, You can gaurantee all numbers from the primorial plus or minus 2, until the primorial plus or minus the first prime not in the primorial minus or plus 1, are composite for 30=2*3*5 you get that all numbers in ranges 24-28 and 32-36 are necessarily composite (divisible by a prime in the factorization of 30). Unsolved conjectures, put bounds on d for all k values. Goldbach, has Bertrand's postulate as a necessary condition. Legendre, implies that two primes exists between $n^2$ and $(n+2)^2$ , n, a natural number. Grimm's, implies that $d<pi(n)$ for k=1, for almost all ( all but finitely many) n. If not then we have a pigeonhole contradiction.

share|cite|improve this answer

edited Mar 14 at 14:20

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116

$endgroup$

add a comment | 

$begingroup$

Which other unsolved problems, have necessary restrictions on the prime gaps? a related question I just got answered. As the comments on your question talk about though, there's not really a restriction. Primorials (products of all primes up to a number) have potentially massive gaps nearby, You can gaurantee all numbers from the primorial plus or minus 2, until the primorial plus or minus the first prime not in the primorial minus or plus 1, are composite for 30=2*3*5 you get that all numbers in ranges 24-28 and 32-36 are necessarily composite (divisible by a prime in the factorization of 30). Unsolved conjectures, put bounds on d for all k values. Goldbach, has Bertrand's postulate as a necessary condition. Legendre, implies that two primes exists between $n^2$ and $(n+2)^2$ , n, a natural number. Grimm's, implies that $d<pi(n)$ for k=1, for almost all ( all but finitely many) n. If not then we have a pigeonhole contradiction.

share|cite|improve this answer

edited Mar 14 at 14:20

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116

$endgroup$

share|cite|improve this answer

edited Mar 14 at 14:20

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116

answered Mar 13 at 21:52

Roddy MacPheeRoddy MacPhee

364116