{"version":"1.0","title":"n!\u304cp\u3067\u5272\u308a\u5207\u308c\u308b\u500b\u6570","published":"2022-06-21 22:09:16","blog_title":"shakayami\u306e\u65e5\u8a18","description":"\u6982\u8981 p\u3092\u7d20\u6570\u3068\u3057\u305f\u3068\u304d\u3001 n!\u304cp\u3067\u5272\u308a\u5207\u308c\u308b\u500b\u6570\u306f $$v_p(n!)=\\dfrac{n-S_p(n)}{p-1}$$ \u3067\u3042\u308b\u3002\u3053\u3053\u3067\u3001$S_p(n)$\u3068\u306f\u3001$n$\u3092$p$\u9032\u6570\u8868\u8a18\u3057\u305f\u3068\u304d\u306e\u5404\u6841\u306e\u7dcf\u548c\u3067\u3042\u308b\u3002 \u8a3c\u660e \u5e30\u7d0d\u6cd5\u3067\u3084\u308b\u3002\u3068\u308a\u3042\u3048\u305a\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u306e\u5b9a\u7406\u306f\u65e2\u77e5\u3068\u3059\u308b\u3002 \u77e5\u3089\u306a\u3044\u5834\u5408\u306f\u9ad8\u6821\u6570\u5b66\u306e\u7f8e\u3057\u3044\u7269\u8a9e\u3092\u898b\u3088\u3046 \u3064\u307e\u308a\u3001 $$v_p(n!)=\\sum_{k=1}^{\\infty}\\lfloor\\dfrac{n}{p^k}\\rfloor$$ \u4efb\u610f\u306e\u6574\u6570$n$\u306f\u3001\u6574\u6570$m $\u30680\u4ee5\u4e0a$p$\u672a\u6e80\u306e\u6574\u6570$a$\u3092\u7528\u3044\u3066 $$n=mp+a$$ \u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002 \u3053\u306e\u3068\u304d\u3001 $$v_p(n!)=\u2026","width":"100%","blog_url":"https://shakayami.hatenablog.com/","author_name":"shakayami","categories":[],"html":"<iframe src=\"https://hatenablog-parts.com/embed?url=https%3A%2F%2Fshakayami.hatenablog.com%2Fentry%2F2022%2F06%2F21%2F220916\" title=\"n!\u304cp\u3067\u5272\u308a\u5207\u308c\u308b\u500b\u6570 - shakayami\u306e\u65e5\u8a18\" class=\"embed-card embed-blogcard\" scrolling=\"no\" frameborder=\"0\" style=\"display: block; width: 100%; height: 190px; max-width: 500px; margin: 10px 0px;\"></iframe>","type":"rich","author_url":"https://blog.hatena.ne.jp/shakayami/","image_url":null,"height":"190","provider_name":"Hatena Blog","provider_url":"https://hatena.blog","url":"https://shakayami.hatenablog.com/entry/2022/06/21/220916"}