Ed Pegg jr.先生发现上图中的线段d长度为
,非常接近7(数值为7.0000000857)[1]
在趣味数学中,接近整数是指很接近整数的无理数。这类数字中,有些因为其数学上的特性使其接近整数,有些还找不到其特性,看起来似乎只是巧合。
有关黄金比例及其他皮索特-维贾亚拉加文数[编辑]
黄金比例
的高次方符合此特性。例如
![{\displaystyle \varphi ^{17}={\frac {3571+1597{\sqrt {5}}}{2}}\approx 3571.00028\approx F_{16}+F_{18}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc251682659e3d34d4f3644bbb79feedd1d8f015)
![{\displaystyle \varphi ^{18}=2889+1292{\sqrt {5}}\approx 5777.999827\approx F_{17}+F_{19}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f52fb14dfa55f21ed348940426cb1ddd6c397067)
![{\displaystyle \varphi ^{19}={\frac {9349+4181{\sqrt {5}}}{2}}\approx 9349.000107\approx F_{18}+F_{20}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d32c25838f05ee822b1139e6ccf962c08c6a674c)
- 其中
代表费波纳契数列的第
项
这是因为有恒等式
[注 1],所以当
为足够大的正整数时,
![{\displaystyle \varphi ^{n}=F_{n-1}+F_{n}\times \varphi \approx F_{n-1}+F_{n}\times \left({\frac {F_{n+1}}{F_{n}}}\right)=F_{n-1}+F_{n+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c97d2ad861b3b878b288b13215fa944ed10277e)
这些数字接近整数的原因和黄金比例的特性有关,不是数学巧合。其原因是因为黄金比例为皮索特-维贾亚拉加文数,而皮索特-维贾亚拉加文数的高次方会是接近整数。
这些数字与费波纳契数有密切的关系,因为费波纳契数相邻两项的比值会趋近于黄金比例,而如果m整除n,则第m个费波纳契数也会整除第n个费波纳契数。
皮索特-维贾亚拉加文数是指代数数本身大于1,而且其极小多项式中另一根的绝对值小于1。像黄金比例本身大于1,
的最小多项式为
另一根为
绝对值小于1,因此黄金比例为皮索特-维贾亚拉加文数,其高次方会是接近整数。
依照根和系数的关系,可得知
而
可以用
及
来表示,由于二根之和及二根之积均为整数,计算所得的结果也是一个正整数,假设为一正整数K,则
可以用下式表示
由于
的绝对值小于1,在n增大时,其高次方会趋于0,此时可得
除了黄金比例外,其他皮索特-维贾亚拉加文数的无理数也符合此一条件,例如
。
有关黑格纳数[编辑]
以下也是几个非巧合出现的接近整数,和最大三项的黑格纳数有关:
![{\displaystyle e^{\pi {\sqrt {43}}}\approx 884736743.999777466\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16dae694978497d411c579c9138c8f1110c27f90)
![{\displaystyle e^{\pi {\sqrt {67}}}\approx 147197952743.999998662454\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c891933d15c8310d5d61f99302b544672645ccc2)
![{\displaystyle e^{\pi {\sqrt {163}}}\approx 262537412640768743.99999999999925007\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5abac9a1b78fafe50650166facb16449cb8f21f)
以上三式可以用以下的式子表示[2]:
![{\displaystyle e^{\pi {\sqrt {43}}}=12^{3}(9^{2}-1)^{3}+744-2.225\cdots \times 10^{-4}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f840aa0917c15ba5c29f0f384719a97f050130d)
![{\displaystyle e^{\pi {\sqrt {67}}}=12^{3}(21^{2}-1)^{3}+744-1.337\cdots \times 10^{-6}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/836ce06685354009bee63cb6b93b39bf36ed4eae)
![{\displaystyle e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+744-7.499\cdots \times 10^{-13}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/330639dff18debe7a194e85a6e79f49df1083ebe)
其中:
由于艾森斯坦级数的关系,使得上式中出现平方项。常数
有时会称为拉马努金常数。
有关π及e[编辑]
许多有关π及e的常数也是接近整数,例如
![{\displaystyle e^{\pi }-\pi =19.999099979189\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc53960822c4a85f29c54e1cb9b0966692514bf)
以及
![{\displaystyle e^{5\pi }=6635623.999341134233\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5c49be8fb5dad079072c0691c0306eb1dba49f)
格尔丰德常数(
)接近
,至2011年为止还没找到出现此特性的原因[1],因此只能视为一数学巧合。另一个有关格尔丰德常数的常数也是接近整数
以下也是一些接近整数的例子
![{\displaystyle 22{\pi }^{4}=2143.0000027480\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b05edcfb361e891890c72cd390acbb8d74c44867)
![{\displaystyle {\pi }^{5}=306.019684\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a084c5d46bdcc60c404310dfe404ec33cc7c706)
![{\displaystyle {\pi }^{3}=31.006276\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2690c836b16a533de61125c08884e77f2bd22a74)
![{\displaystyle {\pi }^{13/2}=1704.017978\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/623db1ef0e3795173438fca96f55238b1a7cad68)
![{\displaystyle {\pi }^{3}-{\frac {\pi }{500}}=30.999993494\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2da859d40a55e86544d043ea7fa65527278bc0d)
![{\displaystyle {\pi }^{2}+{\frac {\pi }{24}}=10.000504\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fdfc5e6f59d2b64a382ce2a619e62ee4b2c830e)
![{\displaystyle {\pi }^{5}-3{\pi }^{3}=213.0008547\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b388eaec54085b0d9605b0e5003dbb76ed0b858)
![{\displaystyle e^{\pi }-\pi +0.0009=19.9999999791\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1da6855dfbd901dcccd6a1a53a8ae3da95c6dc8)
![{\displaystyle e^{3}=20.0855369\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e74a988d07d3c2f4dcf31088a2a3d6b77a536344)
![{\displaystyle e^{9/2}=90.0171313\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1bc5839cebcddf75e94cf30ceb5974a2df14b78)
![{\displaystyle e^{\pi {\sqrt {2}}}=85.019695\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96f358c3b0cec7844d3bec3f50167744d795d89b)
![{\displaystyle e^{6}-{\pi }^{5}-{\pi }^{4}=0.00001767\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61fe1d8e375ae94a0efba98922d8546a5483724a)
![{\displaystyle 9{\pi }^{5}-2e^{3}=2714.00608922\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d583dc5287e3407f7e16bb10fcf6d55989b8089)
![{\displaystyle e^{13/2}-\pi =662.00004039\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b1503e49e4ba1fa5f1c5cf1ee782bb85298278)
![{\displaystyle {\pi }^{13/2}-e^{9/2}=1614.00084707\cdots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42bfad44e8f0ff027155f5cc0d28a1b322047278)
其他例子[编辑]
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![{\displaystyle 2^{2^{2/3}}\approx 3.00507511272}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6200b61dd3bc21c14c2c70fecd59ddda6b7942)
![{\displaystyle 4\ln 2.117\approx 2.99999996861}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daa4a53fc07c283354d50b3634dd8c0c04bdb29)
,其中
是辛钦常数
![{\displaystyle {}_{{\frac {10}{81}}-\sum _{n=1}^{\infty }{\frac {\sum _{k=10^{n-1}}^{10^{n}-1}10^{-n\left[k-(10^{n-1}-1)\right]}k}{10^{\sum _{k=0}^{n-1}9\times 10^{k-1}k}}}={\frac {10}{81}}-\sum _{n=1}^{\infty }\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{k=0}^{n-1}10^{k}(n-k)}}}\approx 1.022344\times 10^{-9}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fce0aa2270420a60e547b6a7d8d260c731e04a9)
![{\displaystyle {}_{-{\frac {1}{5}}+e^{\frac {6}{5}}{}_{4}F_{3}\left(-{\frac {1}{5}},{\frac {1}{20}},{\frac {3}{10}},{\frac {11}{20}};{\frac {1}{5}},{\frac {2}{5}},{\frac {3}{5}};{\frac {256}{3125e^{6}}}\right)+{\frac {2}{25e^{\frac {6}{5}}}}{}_{4}F_{3}\left({\frac {1}{5}},{\frac {9}{20}},{\frac {7}{10}},{\frac {19}{20}};{\frac {3}{5}},{\frac {4}{5}},{\frac {7}{5}};{\frac {256}{3125e^{6}}}\right)-{\frac {4}{125e^{\frac {12}{5}}}}{}_{4}F_{3}\left({\frac {2}{5}},{\frac {13}{20}},{\frac {9}{10}},{\frac {23}{20}};{\frac {4}{5}},{\frac {6}{5}},{\frac {8}{5}};{\frac {256}{3125e^{6}}}\right)+{\frac {7}{625e^{\frac {18}{5}}}}{}_{4}F_{3}\left({\frac {3}{5}},{\frac {17}{20}},{\frac {11}{10}},{\frac {27}{20}};{\frac {6}{5}},{\frac {7}{5}},{\frac {9}{5}};{\frac {256}{3125e^{6}}}\right)-\pi \approx 2.89221114964408683\times 10^{-8}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be2d7cea7422eb58e3edea8fe6bad6f9ffcf1118)
![{\displaystyle {}_{\qquad {\mbox{Root of }}x^{6}-615x^{5}+151290x^{4}-18608670x^{3}+1144433205x^{2}-28153057165x+39605=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1ffee94aabf46aa027d60ef4b859a8ea5f24b4)
![{\displaystyle {}_{{\frac {615-55{\sqrt {5}}-{\sqrt[{3}]{7451370+3332354{\sqrt {5}}+6{\sqrt {8890710030+3976046490{\sqrt {5}}}}}}-{\sqrt[{3}]{7451370+3332354{\sqrt {5}}-6{\sqrt {8890710030+3976046490{\sqrt {5}}}}}}}{6}}\approx 1.40677447684\times 10^{-6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3089725651ebd292f2f5c48ca8cd4dbf48e79ea1)
![{\displaystyle {}_{\qquad {\mbox{Root of }}312500000x^{5}-6843750000x^{4}+6826250000x^{3}+10476025000x^{2}-7886869750x-72099=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2d1549aeadec338793842e0e23ea015ad1878c)
![{\displaystyle {}_{\tan \left({\frac {\arctan 4}{5}}+{\frac {4\pi }{5}}\right)+{\frac {19}{50}}={\frac {219}{50}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{884+799{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{884-799{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{1156+289{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{1156-289{\rm {i}}}}\approx -9.141538637378949398666277\times 10^{-6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5eccb514738b7a3041bd62704e2a92e9376fa29)
![{\displaystyle {}_{\rm {{erfi}\left({\rm {{erfi}{\frac {\sqrt {3}}{3}}}}\right)={\frac {2}{\sqrt {\pi }}}\int _{0}^{{\frac {2}{\sqrt {\pi }}}\int _{0}^{\frac {\sqrt {3}}{3}}e^{t^{2}}{\rm {{d}t}}}e^{u^{2}}{\rm {{d}u={\frac {2}{\sqrt {\pi }}}e^{\left({\frac {2{\sqrt[{3}]{e}}}{\sqrt {\pi }}}\int _{0}^{\infty }{\frac {\sin \left({\frac {2}{3}}{\sqrt {3}}t\right)}{e^{t^{2}}}}{\rm {d}}t\right)^{2}}\int _{0}^{\infty }{\frac {\sin \left[{\frac {4u{\sqrt[{3}]{e}}}{\sqrt {\pi }}}\int _{0}^{\infty }{\frac {\sin \left({\frac {2}{3}}{\sqrt {3}}t\right)}{e^{t^{2}}}}{\rm {d}}t\right]}{e^{u^{2}}}}{\rm {d}}u={\frac {2}{\sqrt {\pi }}}\int _{0}^{{}_{{\frac {2{\sqrt[{3}]{e}}}{\sqrt {\pi }}}\int _{0}^{\infty }{\frac {\sin \left({\frac {2}{3}}{\sqrt {3}}t\right)}{e^{t^{2}}}}{\rm {d}}t}}e^{u^{2}}{\rm {d}}u={\frac {2}{\sqrt {\pi }}}e^{\left({\frac {2}{\sqrt {\pi }}}\int _{0}^{\frac {\sqrt {3}}{3}}e^{t^{2}}{\rm {{d}t}}\right)^{2}}\int _{0}^{\infty }{\frac {\sin \left({\frac {4u}{\sqrt {\pi }}}\int _{0}^{\frac {\sqrt {3}}{3}}e^{t^{2}}{\rm {{d}t}}\right)}{e^{u^{2}}}}{\rm {d}}u\approx 1.00002087363809430195879}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be1e0060b8bc6af24aa71f3328cd50f56b4b988)
,这是由于
的缘故,另一个类似的例子为![{\displaystyle \sin 355=-0.00003014435335948844921433...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a237b5e4d70b962cb94db494900cd0d80678823)
![{\displaystyle {\sqrt {1^{2}+2^{2}+3^{2}+4^{2}+.......+552057^{2}}}\approx 236818619.0000004307}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48232ba5481936f21f7f65b711838d255f89f493)
外部链接[编辑]
参考资料[编辑]