{"id":1693,"date":"2025-07-23T23:28:53","date_gmt":"2025-07-23T14:28:53","guid":{"rendered":"https:\/\/math-friend.com\/?p=1693"},"modified":"2025-07-29T11:09:22","modified_gmt":"2025-07-29T02:09:22","slug":"%e3%80%90-%e5%a4%a7%e9%98%aa%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91n%e7%95%aa%e7%9b%ae%e3%81%ae%e7%b4%a0%e6%95%b0%e3%81%ae%e5%a4%a7%e3%81%8d%e3%81%952024","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=1693","title":{"rendered":"\u3010 \u5927\u962a\u5927\u5b66\u5165\u8a66\u3011n\u756a\u76ee\u306e\u7d20\u6570\u306e\u5927\u304d\u3055(2024)"},"content":{"rendered":"\n<p>\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n<p class=\"has-border -border02\">\\(n\\)\u756a\u76ee\u306e\u7d20\u6570\u3092\\(p_n\\)\u3067\u8868\u3059. \u4f8b\u3048\u3070, \\(p_1=2\\), \\(p_2=3\\), \\(p_3=5\\)\u3067\u3042\u308b. <br>\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<br>(1) \\(p_{15}\\)\u3092\u6c42\u3081\u3088.<br>(2) \\(n\\geq 12\\)\u306e\u3068\u304d, \\(p_n&gt;3n\\)\u3092\u793a\u305b.<br><span style=\"text-align:right;display:block;\">(2024 \u5927\u962a\u5927\u5b66 [3])<\/span><\/p>\n\n\n\n<p>\u7d20\u6570\u306e\u5927\u304d\u3055\u3092\u8a55\u4fa1\u3059\u308b\u554f\u984c\u3067\u3059. \u5b9f\u306f\u5927\u5b66\u3067\u5b66\u3076\u7d20\u6570\u5b9a\u7406\u3092\u7528\u3044\u308b\u3053\u3068\u3067, \u4efb\u610f\u306e\u81ea\u7136\u6570\\(k\\)\u306b\u5bfe\u3057\u3066, \u3042\u308b\u6574\u6570\\(N\\)\u304c\u5b58\u5728\u3057\u3066, <br>$$<br>n\\geq N\\,\\,\\Longrightarrow p_n&gt;kn<br>$$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u307e\u3059. \u4eca\u56de\u306e\u554f\u984c\u3067\u306f, \\(k=3\\)\u306e\u3068\u304d\u3067, \u305d\u306e\u3068\u304d\u306e\\(N\\)\u304c\\(12\\)\u3068\u3044\u3046\u3053\u3068\u3067\u3059. \u3053\u306e\u8a71\u306f\u89e3\u7b54\u5f8c\u306b\u304a\u307e\u3051\u3067\u8a18\u8f09\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059.<br><br>\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n<div class=\"wp-block-group is-style-crease\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>(1) \u7d20\u6570\u3092\u5c0f\u3055\u3044\u9806\u306b\u4e26\u3079\u308b\u3068,\\(p_1=2\\), \\(p_2=3\\), \\(p_3=5\\), \\(p_4=7\\), \\(p_5=11\\), \\(p_6=13\\), \\(p_7=17\\), \\(p_8=19\\), \\(p_9=23\\), \\(p_{10}=29\\), \\(p_{11}=31\\), \\(p_{12}=37\\), \\(p_{13}=41\\), \\(p_14=43\\), \\(p_{15}=47\\)\u3068\u306a\u308a,<br>$$<br>p_{15}=47<br>$$\u3067\u3042\u308b.<\/p>\n\n\n\n<p>(2) \\(n\\)\u306b\u95a2\u3059\u308b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u8a3c\u660e\u3059\u308b.<br>\u2460 \\(n=12\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059.<br><br>\u3053\u308c\u306f\\(p_{12}=37 &gt; 36=3\\cdot 12\\)\u3088\u308a\u6210\u308a\u7acb\u3064.<br><br>\u2461 \\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066, \\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059.<br><br>\\(n=k\\)\u306e\u3068\u304d, \u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b. \u3064\u307e\u308a, \\(p_k&gt;3k\\)\u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b\u3068, \\(p_k\\)\u306f\u81ea\u7136\u6570\u3067\u3042\u308b\u304b\u3089, \\(p_k\\geq 3k+1\\)\u3067\u3042\u308b. \u9023\u7d9a\u3059\u308b\u81ea\u7136\u6570\u306f\u7247\u65b9\u304c\u5076\u6570(\\(2\\)\u3067\u5272\u308c\u308b)\u306b\u306a\u308b\u305f\u3081, \\(p_1=2\\), \\(p_2=3\\)\u3092\u9664\u3044\u3066, \u9023\u7d9a\u3059\u308b\u81ea\u7136\u6570\u304c\u7d20\u6570\u306b\u306a\u308b\u3053\u3068\u306f\u306a\u3044\u304b\u3089, \\(p_{k+1}\\geq p_k + 2\\)\u3067\u3042\u308b. \u3088\u3063\u3066, <br>$$<br>p_{k+1}\\geq p_k+2\\geq (3k+1)+2=3(k+1)<br>$$\u3067\u3042\u308b. \u3057\u304b\u3057, \\(3(k+1)\\)\u306f\\(3\\)\u306e\u500d\u6570\u3067\u3042\u308b\u304b\u3089, \\(k=0\\)\u3092\u9664\u3044\u3066\u7d20\u6570\u306b\u306f\u306a\u308a\u5f97\u306a\u3044. \u3088\u3063\u3066\\(p_{k+1}\\neq 3(k+1)\\)\u3068\u306a\u308a, <br>$$<br>p_{k+1}&gt;3(k+1)<br>$$ \u304c\u308f\u304b\u308b.<br><br>\u2460, \u2461\u3088\u308a\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u304b\u3089, \\(n\\geq 12\\)\u306e\u3068\u304d, \\(p_n&gt;3n\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\n<\/div><\/div>\n\n\n\n<p>\u5b9f\u969b\u306b, \u6a2a\u8ef8\u3092\\(n\\)\u3068\u3057, \\(p_n\\), \\(3n\\)\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059. \u305f\u3057\u304b\u306b, \\(n\\geq 12\\)\u306e\u7bc4\u56f2\u3067, \\(p_n&gt;3n\\)\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"989\" height=\"590\" src=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output1.png\" alt=\"\" class=\"wp-image-1714\" style=\"width:503px;height:auto\" srcset=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output1.png 989w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output1-300x179.png 300w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output1-768x458.png 768w\" sizes=\"(max-width: 989px) 100vw, 989px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"989\" height=\"590\" src=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output.png\" alt=\"\" class=\"wp-image-1713\" style=\"width:502px;height:auto\" srcset=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output.png 989w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output-300x179.png 300w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/07\/output-768x458.png 768w\" sizes=\"(max-width: 989px) 100vw, 989px\" \/><\/figure>\n\n\n\n<p class=\"is-style-icon_info\">\u3053\u3053\u304b\u3089\u306f\u5b8c\u5168\u306b\u8da3\u5473\u306e\u4e16\u754c\u306b\u5165\u308a\u307e\u3059. \u53d7\u9a13\u751f\u306f\u8aad\u3080\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093 .<\/p>\n\n\n\n<p>\u4efb\u610f\u306e\u81ea\u7136\u6570\\(k\\)\u306b\u5bfe\u3057\u3066, \u3042\u308b\u6574\u6570\\(N\\)\u304c\u5b58\u5728\u3057\u3066, <br>$$<br>n\\geq N\\,\\,\\Longrightarrow p_n&gt;kn<br>$$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u7d20\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u3056\u3063\u304f\u308a\u3068\u3057\u305f\u8a3c\u660e\u3092\u3057\u3066\u307f\u307e\u3057\u3087\u3046. \u307e\u305a\u7d20\u6570\u5b9a\u7406\u3092\u7d39\u4ecb\u3057\u307e\u3059.<\/p>\n\n\n\n<p class=\"is-style-big_icon_point\">\\(\\pi(x)\\)\u3092\\(x\\)\u4ee5\u4e0b\u306e\u7d20\u6570\u306e\u500b\u6570\u3092\u8fd4\u3059\u95a2\u6570\u3068\u5b9a\u7fa9\u3059\u308b\u3068, \u5341\u5206\u5927\u304d\u3044\u81ea\u7136\u6570\\(n\\)\u306b\u5bfe\u3057\u3066,<br>$$<br>\\pi(x)\\sim \\frac{x}{\\log{x}}<br>$$\u3068\u306a\u308b.<\/p>\n\n\n\n<p>\u305d\u308c\u3067\u306f\u3053\u306e\u7d20\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u3056\u3063\u304f\u308a\u3068\u3057\u305f\u8a3c\u660e\u3092\u3057\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n<p class=\"has-border -border03\">\u81ea\u7136\u6570\\(n\\)\u306b\u5bfe\u3057\u3066\\(\\pi(p_n)=n\\)\u3067\u3042\u308b\u304b\u3089, \u5341\u5206\u5927\u304d\u3044\\(n\\)\u306b\u5bfe\u3057\u3066, \u7d20\u6570\u5b9a\u7406\u304b\u3089,<br>$$<br>n\\sim \\frac{p_n}{\\log{p_n}}<br>$$\u3068\u306a\u308a, <br>$$<br>p_n\\sim n\\log{p_n}<br>$$\u3067\u3042\u308b. \u3053\u3053\u3067, \u660e\u3089\u304b\u306b\\(p_n\\geq n\\)\u3067\u3042\u308b\u304b\u3089, <br>$$<br>p_n\\sim n\\log{p_n} \\geq n\\log{n}<br>$$\u3068\u306a\u308b. \u4efb\u610f\u306e\\(k\\)\u306b\u5bfe\u3057\u3066, \\(n\\)\u3092\u5341\u5206\u5927\u304d\u304f\u3068\u308c\u3070, <br>$$<br>k&lt;\\log{n}<br>$$\u3068\u3067\u304d\u308b\u306e\u3067, \u4ee5\u4e0a\u304b\u3089,<br>$$<br>p_n\\sim n\\log{p_n} \\geq n\\log{n} &gt; kn<br>$$\u3068\u306a\u308b.<\/p>\n\n\n\n<p>\u3053\u3061\u3089\u306e\u8a3c\u660e\u306f\u66d6\u6627\u306a\u8b70\u8ad6\u3082\u3042\u308b\u305f\u3081\u5b8c\u5168\u306a\u8a3c\u660e\u3067\u306f\u306a\u3044\u3053\u3068\u3092\u6ce8\u610f\u3057\u3066\u304a\u304d\u307e\u3059.<\/p>\n\n\n\n<p>youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n<iframe width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/QKEe8OKk_Wk?si=C9UCvC8YcVOIn0Lv\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059. \\(n\\)\u756a\u76ee\u306e\u7d20\u6570\u3092\\(p_n\\)\u3067\u8868\u3059. \u4f8b\u3048\u3070, \\(p_1=2\\), \\(p_2=3\\), \\(p_3=5\\)\u3067\u3042\u308b. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.(1) \\(p_{15 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1711,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[6],"tags":[],"class_list":["post-1693","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tokimakuru"],"_links":{"self":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/1693","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1693"}],"version-history":[{"count":20,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/1693\/revisions"}],"predecessor-version":[{"id":2062,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/1693\/revisions\/2062"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/media\/1711"}],"wp:attachment":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1693"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1693"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1693"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}