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.










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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

















up vote
1
down vote

favorite
2












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.










share|cite|improve this question
























  • 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















up vote
1
down vote

favorite
2









up vote
1
down vote

favorite
2






2





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.










share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Nov 23 at 14:12









Tianlalu

3,0101938




3,0101938










asked Nov 23 at 14:07









China

1,4041029




1,4041029












  • 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










  • @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

















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