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1972.国际奥林匹克IMO.3数论——Legendre定理与高斯函数不等式

2023-08-25 19:34 作者:封包  | 我要投稿

世界的意义必定在世界之外。世界中一切事情就如它们之所是而是，如它们之所发生而发生；世界中不存在价值——如果存在价值，那它也会是无价值的。

——《逻辑哲学论》路德维希‧维特根斯坦

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%7B%5Clarge%20%0A%5Cbegin%7Barray%7D%7Bc%7D%0A%E5%BC%95%E7%90%861%EF%BC%88Legendre%E5%AE%9A%E7%90%86%EF%BC%8C%E5%8B%92%E8%AE%A9%E5%BE%B7%E5%AE%9A%E7%90%86%EF%BC%89%EF%BC%9A%5C%5C%0An!%E4%B8%AD%E4%BB%BB%E6%84%8F%E7%B4%A0%E5%9B%A0%E5%AD%90p%E7%9A%84%E6%AC%A1%E5%B9%82e%E6%BB%A1%E8%B6%B3%E5%85%B3%E7%B3%BB%EF%BC%9A%5C%5C%0Ae%3D%5Csum_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%5B%5Cfrac%7Bn%7D%7Bp%5Ei%7D%20%5D%5C%5C%0A%E8%AF%81%E6%98%8E%E8%AF%A6%E8%A7%81%E3%80%8A%E5%88%9D%E7%AD%89%E6%95%B0%E8%AE%BA%E3%80%8B%E6%95%99%E6%9D%90%EF%BC%8C%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE%E3%80%82%5C%5C%5C%5C%0A%E5%BC%95%E7%90%862%EF%BC%88%E9%AB%98%E6%96%AF%E5%87%BD%E6%95%B0%EF%BC%89%3A%0A%E5%AF%B9%5Cforall%20a%2Cb%5Cin%20%5Cmathbb%7BR%7D%3A%5C%5C%0A%20%5Ba%5D%2B%5Bb%5D%2B%5Ba%2Bb%5D%E2%89%A4%5B2a%5D%2B%5B2b%5D%5C%5C%0A%E5%BC%95%E7%90%862%E8%AF%81%E6%98%8E%EF%BC%9A%5C%5C%0A%E5%AE%9A%E4%B9%89%5C%7Ba%5C%7D%3Da-%5Ba%5D%5C%5C%0AI.%E8%8B%A5%5C%7Ba%5C%7D%2C%5C%7Bb%5C%7D%E2%89%A5%5Cfrac%7B1%7D%7B2%7D%3A%5C%5C%0ARHS%3D2%5Ba%5D%2B2%5Bb%5D%2B2%5C%5C%0ALHS%3D%5Ba%5D%2B%5Bb%5D%2B%5Ba%5D%2B%5Bb%5D%2B1%20%5C%5C%0A%E2%88%B4RHS%EF%BC%9ELHS%5C%5C%0AII.%E8%8B%A5%5C%7Ba%5C%7D%2C%5C%7Bb%5C%7D%3C%5Cfrac%7B1%7D%7B2%7D%3A%5C%5C%0ARHS%3D2%5Ba%5D%2B2%5Bb%5D%3DLHS%0AIII.%E8%8B%A51%3E%5C%7Ba%5C%7D%E2%89%A5%5Cfrac%7B1%7D%7B2%7D%3E%5C%7Bb%5C%7D%3A%5C%5C%0ARHS%3D2%5Ba%5D%2B2%5Bb%5D%2B1%5C%5C%0ALHS%E2%89%A42%5Ba%5D%2B2%5Bb%5D%2B1%5C%5C%0A%E2%88%B4RHS%E2%89%A5LHS%5C%5C%0A(%E7%94%B1%E5%AF%B9%E7%A7%B0%E6%80%A7%E7%9F%A5%E5%8F%A6%E4%B8%80%E9%83%A8%E5%88%86%E6%88%90%E7%AB%8B)%5C%5C%0A%E7%BB%BC%E4%B8%8A%E5%BC%95%E7%90%862%E5%BE%97%E8%AF%81%E3%80%82%5C%5C%5C%5C%0A%E7%8E%B0%E5%9C%A8%E6%9D%A5%E8%AF%81%E6%98%8E%E6%88%91%E4%BB%AC%E7%9A%84%E5%8E%9F%E5%91%BD%E9%A2%98%EF%BC%9A%5C%5C%0A%E5%8E%9F%E5%91%BD%E9%A2%98%E8%BD%AC%E5%8C%96%E4%B8%BA%3A%5C%5C%0Am!n!(m%2Bn)!%7C(2m)!(2n)!%5C%5C%0A%E6%9E%84%E9%80%A0%E5%87%BD%E6%95%B0V_%7Bp_i%7D(x)%E4%B8%BAx%E6%A0%87%E5%87%86%E5%88%86%E8%A7%A3%E5%BC%8F%E4%B8%ADp_i%E7%9A%84%E6%AC%A1%E5%B9%82%E5%A4%A7%E5%B0%8F%E3%80%82%5C%5C%0Ap_i%E4%B8%BAx%E7%B4%A0%E5%9B%A0%E5%AD%90%5C%5C%20%0A%E5%88%99%E5%8E%9F%E5%91%BD%E9%A2%98%E8%BD%AC%E5%8C%96%E4%B8%BA%3A%5C%5C%0A%5Cforall%20p_i%2CV_%7Bp_i%7D(m!n!(m%2Bn)!)%3D%5C%5C%0AV_%7Bp_i%7D(m!)%2BV_%7Bp_i%7D(n!)%2BV_%7Bp_i%7D((m%2Bn)!)%5C%5C%0A%E2%89%A4V_%7Bp_i%7D((2m)!(2n)!)%3DV_%7Bp_i%7D((2m)!)%2BV_%7Bp_i%7D((2n)!)%5C%5C%0A%E7%94%B1%E5%BC%95%E7%90%861(Legendre%E5%AE%9A%E7%90%86)%3A%5C%5C%0A%E5%8D%B3%E8%AF%81%3A%5Csum%5Cnolimits_%7Bi%3D1%7D%5E%7B%5Cinfty%20%7D%20%5B%5Cfrac%7Bm%7D%7Bp%5Ei%7D%5D%2B%5B%5Cfrac%7Bn%7D%7Bp%5Ei%7D%5D%2B%5B%5Cfrac%7Bn%2Bm%7D%7Bp%5Ei%7D%5D%20%E2%89%A4%0A%5Csum%5Cnolimits_%7Bi%3D1%7D%5E%7B%5Cinfty%20%7D%5B%5Cfrac%7B2m%7D%7Bp%5Ei%7D%5D%2B%5B%5Cfrac%7B2n%7D%7Bp%5Ei%7D%5D%5C%5C%0A%E7%94%B1%E5%BC%95%E7%90%862%E7%9F%A5%3A%5C%5C%0A%5B%5Cfrac%7Bm%7D%7Bp%5Ei%7D%5D%2B%5B%5Cfrac%7Bn%7D%7Bp%5Ei%7D%5D%2B%5B%5Cfrac%7Bn%2Bm%7D%7Bp%5Ei%7D%5D%E2%89%A4%5B%5Cfrac%7B2m%7D%7Bp%5Ei%7D%5D%2B%5B%5Cfrac%7B2n%7D%7Bp%5Ei%7D%5D%E6%81%92%E6%88%90%E7%AB%8B%5C%5C%0A%E6%89%80%E4%BB%A5%E4%B8%8A%E8%BF%B0%E5%BC%8F%E5%AD%90%E6%81%92%E6%88%90%E7%AB%8B%EF%BC%8C%E5%91%BD%E9%A2%98%E6%88%90%E7%AB%8B%E3%80%82%5C%5C%0A%E8%AF%81%E6%AF%95%E3%80%82%5C%5C%0A%5Cend%7Barray%7D%7D%20

勒让德定理是数论中常见且基本的一个定理，一般在学习素数基本定理的时候会补充。本题思路并无难处，但是步骤较为繁琐（也指我码字繁琐），能够熟悉高斯函数的性质是关键。

高中生自行整理，如有错误，欢迎指出。

标签：数学

1972.国际奥林匹克IMO.3数论——Legendre定理与高斯函数不等式的评论 (共 条)

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