About the sum of the reciprocals of the twin primes
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The sum of the reciprocals of the primes verifying $p_{n+1}=p_{n}+2$ (pairs of primes which differ by $2$) converges to a finite value
$$B_2=sum_{n} frac1{p_{n}}+frac1{p_{n}+2}>0$$
In other words, the sum either has finitely many terms or has infinitely many terms but is convergent. In 2004 Thomas Nicely gave $$B_2=1.9021605825820±000000001620$$ based on all twin primes less than $5×10^{15}$.
My question is: How one can prove rigorously that $B_2<2$. The estimation of Nicely is not a mathematical proof.
number-theory elementary-number-theory
edited Nov 23 at 14:12
Tianlalu
3,0101938
asked Nov 23 at 14:07
China
1,4021029
add a comment |
up vote
1
down vote
favorite
The sum of the reciprocals of the primes verifying $p_{n+1}=p_{n}+2$ (pairs of primes which differ by $2$) converges to a finite value
$$B_2=sum_{n} frac1{p_{n}}+frac1{p_{n}+2}>0$$
In other words, the sum either has finitely many terms or has infinitely many terms but is convergent. In 2004 Thomas Nicely gave $$B_2=1.9021605825820±000000001620$$ based on all twin primes less than $5×10^{15}$.
My question is: How one can prove rigorously that $B_2<2$. The estimation of Nicely is not a mathematical proof.
number-theory elementary-number-theory
edited Nov 23 at 14:12
Tianlalu
3,0101938
asked Nov 23 at 14:07
China
1,4021029
this claims to be a proof (I haven't gone over it carefully).
– lulu
Nov 23 at 14:16
@lulu: I am interested on the location of $B2$ not on the proof of the Brun theorem.
– China
Nov 23 at 14:19
Effective version of Brun's theorem are bounds on $sum_{n > x} frac{1_{p_n+2 in P}}{p_n}$
– reuns
Nov 23 at 19:12
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The sum of the reciprocals of the primes verifying $p_{n+1}=p_{n}+2$ (pairs of primes which differ by $2$) converges to a finite value
$$B_2=sum_{n} frac1{p_{n}}+frac1{p_{n}+2}>0$$
In other words, the sum either has finitely many terms or has infinitely many terms but is convergent. In 2004 Thomas Nicely gave $$B_2=1.9021605825820±000000001620$$ based on all twin primes less than $5×10^{15}$.
My question is: How one can prove rigorously that $B_2<2$. The estimation of Nicely is not a mathematical proof.
number-theory elementary-number-theory
edited Nov 23 at 14:12
Tianlalu
3,0101938
asked Nov 23 at 14:07
China
1,4021029
The sum of the reciprocals of the primes verifying $p_{n+1}=p_{n}+2$ (pairs of primes which differ by $2$) converges to a finite value
$$B_2=sum_{n} frac1{p_{n}}+frac1{p_{n}+2}>0$$
In other words, the sum either has finitely many terms or has infinitely many terms but is convergent. In 2004 Thomas Nicely gave $$B_2=1.9021605825820±000000001620$$ based on all twin primes less than $5×10^{15}$.
My question is: How one can prove rigorously that $B_2<2$. The estimation of Nicely is not a mathematical proof.
number-theory elementary-number-theory
number-theory elementary-number-theory
edited Nov 23 at 14:12
Tianlalu
3,0101938
asked Nov 23 at 14:07
China
1,4021029
edited Nov 23 at 14:12
Tianlalu
3,0101938
asked Nov 23 at 14:07
China
1,4021029
edited Nov 23 at 14:12
Tianlalu
3,0101938
edited Nov 23 at 14:12
Tianlalu
3,0101938
edited Nov 23 at 14:12
Tianlalu
3,0101938
3,0101938
asked Nov 23 at 14:07
China
1,4021029
asked Nov 23 at 14:07
China
1,4021029
asked Nov 23 at 14:07
China
1,4021029
1,4021029
this claims to be a proof (I haven't gone over it carefully).
– lulu
Nov 23 at 14:16
@lulu: I am interested on the location of $B2$ not on the proof of the Brun theorem.
– China
Nov 23 at 14:19
Effective version of Brun's theorem are bounds on $sum_{n > x} frac{1_{p_n+2 in P}}{p_n}$
– reuns
Nov 23 at 19:12
add a comment |
this claims to be a proof (I haven't gone over it carefully).
– lulu
Nov 23 at 14:16
@lulu: I am interested on the location of $B2$ not on the proof of the Brun theorem.
– China
Nov 23 at 14:19
Effective version of Brun's theorem are bounds on $sum_{n > x} frac{1_{p_n+2 in P}}{p_n}$
– reuns
Nov 23 at 19:12
this claims to be a proof (I haven't gone over it carefully).
– lulu
Nov 23 at 14:16
this claims to be a proof (I haven't gone over it carefully).
– lulu
Nov 23 at 14:16
@lulu: I am interested on the location of $B2$ not on the proof of the Brun theorem.
– China
Nov 23 at 14:19
@lulu: I am interested on the location of $B2$ not on the proof of the Brun theorem.
– China
Nov 23 at 14:19
Effective version of Brun's theorem are bounds on $sum_{n > x} frac{1_{p_n+2 in P}}{p_n}$
– reuns
Nov 23 at 19:12
Effective version of Brun's theorem are bounds on $sum_{n > x} frac{1_{p_n+2 in P}}{p_n}$
– reuns
Nov 23 at 19:12
add a comment |
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this claims to be a proof (I haven't gone over it carefully).
– lulu
Nov 23 at 14:16
@lulu: I am interested on the location of $B2$ not on the proof of the Brun theorem.
– China
Nov 23 at 14:19
Effective version of Brun's theorem are bounds on $sum_{n > x} frac{1_{p_n+2 in P}}{p_n}$
– reuns
Nov 23 at 19:12