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孿生質數者，質數之差二也，若三與五，五與七爾爾。
3, 5, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, etc
孿生質數之無窮乎。遂有數學家究之，然求其倒數之和，值收斂，名曰布隆常數[一]
1 3 + 1 5 + 1 5 + 1 7 + 1 11 + 1 13 + ⋯ ⋯ = 1.90216058 ⋯ ⋯ {\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+\cdots \cdots =1.90216058\cdots \cdots }
今仍無解，或下文爲可證其無窮性。
全部證明請見條目之孪生质数无穷多的证明者也.
N(twin primes)
= lim n → ∞ [ n ⋅ 1 3 ( 1 − 2 5 ) ( 1 − 2 7 ) ( 1 − 2 11 ) ( 1 − 2 13 ) . . . . . . ] {\displaystyle \mathrm {=\lim _{n\to \infty }[n\cdot {\frac {1}{3}}(1-{\frac {2}{5}})(1-{\frac {2}{7}})(1-{\frac {2}{11}})(1-{\frac {2}{13}})......]} }
= lim n → ∞ [ n ⋅ ( 1 − 2 3 ) ( 1 − 2 5 ) ( 1 − 2 7 ) ( 1 − 2 11 ) ( 1 − 2 13 ) . . . . . . ] {\displaystyle \mathrm {=\lim _{n\to \infty }[n\cdot (1-{\frac {2}{3}})(1-{\frac {2}{5}})(1-{\frac {2}{7}})(1-{\frac {2}{11}})(1-{\frac {2}{13}})......]} }
= lim n → ∞ n ⋅ k [ ( 1 − 1 3 ) ( 1 − 1 5 ) ( 1 − 1 7 ) ( 1 − 1 11 ) ( 1 − 1 13 ) . . . . . . ] 2 {\displaystyle \mathrm {=\lim _{n\to \infty }n\cdot k[(1-{\frac {1}{3}})(1-{\frac {1}{5}})(1-{\frac {1}{7}})(1-{\frac {1}{11}})(1-{\frac {1}{13}})......]^{2}} }
= lim n → ∞ 4 k n [ ( 1 − 1 2 ) ( 1 − 1 3 ) ( 1 − 1 5 ) ( 1 − 1 7 ) ( 1 − 1 11 ) ( 1 − 1 13 ) . . . . . . ] 2 {\displaystyle \mathrm {=\lim _{n\to \infty }4kn[(1-{\frac {1}{2}})(1-{\frac {1}{3}})(1-{\frac {1}{5}})(1-{\frac {1}{7}})(1-{\frac {1}{11}})(1-{\frac {1}{13}})......]^{2}} }
= lim n → ∞ 4 k n [ ln ( n ) + γ ] 2 {\displaystyle \mathrm {=\lim _{n\to \infty }{\frac {4kn}{[\ln(n)+\gamma ]^{2}}}} }
= ∞ {\displaystyle \mathrm {=\infty } }
![{\displaystyle \mathrm {=\lim _{n\to \infty }[n\cdot {\frac {1}{3}}(1-{\frac {2}{5}})(1-{\frac {2}{7}})(1-{\frac {2}{11}})(1-{\frac {2}{13}})......]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d77b49eb3224d9eee9f7d4c32d7c2df330445e59)
![{\displaystyle \mathrm {=\lim _{n\to \infty }[n\cdot (1-{\frac {2}{3}})(1-{\frac {2}{5}})(1-{\frac {2}{7}})(1-{\frac {2}{11}})(1-{\frac {2}{13}})......]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/eefc272f349ed977525df66d8260cb04f5249cd9)
![{\displaystyle \mathrm {=\lim _{n\to \infty }n\cdot k[(1-{\frac {1}{3}})(1-{\frac {1}{5}})(1-{\frac {1}{7}})(1-{\frac {1}{11}})(1-{\frac {1}{13}})......]^{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3015c0541582d07ae3b78c79d07d917e8becc3)
![{\displaystyle \mathrm {=\lim _{n\to \infty }4kn[(1-{\frac {1}{2}})(1-{\frac {1}{3}})(1-{\frac {1}{5}})(1-{\frac {1}{7}})(1-{\frac {1}{11}})(1-{\frac {1}{13}})......]^{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa9e192d7fbc67267f7fc6c377fce6a6236e2ae)
![{\displaystyle \mathrm {=\lim _{n\to \infty }{\frac {4kn}{[\ln(n)+\gamma ]^{2}}}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0885632a2ecb538e04f4922d006e421a01289457)
