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 難題 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。 Remove ads無窮性之證 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 } } Remove ads兼查 senary duodecimal 據Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads
![{\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}})......]} }](//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}})......]} }](//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}} }](//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}} }](//wikimedia.org/api/rest_v1/media/math/render/svg/2aa9e192d7fbc67267f7fc6c377fce6a6236e2ae)
![{\displaystyle \mathrm {=\lim _{n\to \infty }{\frac {4kn}{[\ln(n)+\gamma ]^{2}}}} }](//wikimedia.org/api/rest_v1/media/math/render/svg/0885632a2ecb538e04f4922d006e421a01289457)
![{\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}})......]} }](http://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}})......]} }](http://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}} }](http://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}} }](http://wikimedia.org/api/rest_v1/media/math/render/svg/2aa9e192d7fbc67267f7fc6c377fce6a6236e2ae)
![{\displaystyle \mathrm {=\lim _{n\to \infty }{\frac {4kn}{[\ln(n)+\gamma ]^{2}}}} }](http://wikimedia.org/api/rest_v1/media/math/render/svg/0885632a2ecb538e04f4922d006e421a01289457)
