{"id":52,"date":"2025-06-29T01:24:37","date_gmt":"2025-06-28T16:24:37","guid":{"rendered":"https:\/\/math-friend.com\/?p=52"},"modified":"2025-07-29T10:17:20","modified_gmt":"2025-07-29T01:17:20","slug":"%e4%ba%ac%e9%83%bd%e5%a4%a7%e5%ad%a6-%e6%9c%89%e5%90%8d%e3%81%aa%e7%b4%a0%e6%95%b0%e3%82%92%e7%94%a8%e3%81%84%e3%81%9f%e5%95%8f%e9%a1%8c","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=52","title":{"rendered":"\u3010\u4eac\u90fd\u5927\u5b66\u5165\u8a66\u3011\u7d20\u6570\u304c\u51fa\u3066\u304f\u308b\u6709\u540d\u554f\u984c(2016)"},"content":{"rendered":"\n<p class=\"u-mb-ctrl u-mb-10\">\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059.<\/p>\n\n\n\n<p class=\"has-border -border02\">\\(p\\), \\(q\\)\u3092\u7d20\u6570\u3068\u3059\u308b\u3068\u304d, \\(p^q+q^p\\)\u306e\u5f62\u3067\u8868\u305b\u308b\u7d20\u6570\u3092\u5168\u3066\u6c42\u3081\u3088. (2016 \u4eac\u90fd\u5927\u5b66)<\/p>\n\n\n\n<p>\u3053\u308c\u306f2016\u5e74\u306e\u4eac\u90fd\u5927\u5b66\u306e\u5165\u8a66\u554f\u984c\u3067\u51fa\u984c\u3055\u308c\u305f\u6709\u540d\u554f\u984c\u3067\u3059. <br><br>\u7d20\u6570\u306f\u672a\u3060\u306b\u8b0e\u304c\u591a\u304f, \u591a\u304f\u306e\u6570\u5b66\u8005\u304c\u305d\u306e\u8b0e\u306e\u89e3\u660e\u306e\u305f\u3081\u306b\u7814\u7a76\u3092\u3057\u3066\u3044\u307e\u3059. \u3042\u308b\u5f62\u3092\u3057\u305f\u7d20\u6570\u304c\u7121\u9650\u306b\u3042\u308b\u304b\u3069\u3046\u304b\u3082\u307b\u3068\u3093\u3069\u77e5\u3089\u308c\u3066\u3044\u307e\u305b\u3093. \u4f8b\u3048\u3070\\(2^n+1\\) , \\(2^n-1\\), \\(n^2+1\\)\u306e\u3088\u3046\u306b\u5272\u3068\u7c21\u5358\u306a\u5f62\u3092\u3057\u305f\u7d20\u6570\u3067\u3059\u3089\u7121\u9650\u306b\u3042\u308b\u304b\u3069\u3046\u304b\u304c\u308f\u304b\u3063\u3066\u3044\u306a\u3044\u306e\u3067\u3059. \u5c11\u3057\u8003\u3048\u308c\u3070\u308f\u304b\u308a\u307e\u3059\u304c, \\(n^2-1\\)\u306e\u5f62\u3092\u3057\u305f\u7d20\u6570\u306f\\(3\\)\u3060\u3051\u3067\u3042\u308b\u3053\u3068\u306f\u77e5\u3089\u308c\u3066\u3044\u307e\u3059. <br><br>\u3053\u306e\u3088\u3046\u306b\u8b0e\u591a\u304d\u7d20\u6570\u3067\u3059\u304b\u3089, \u4eca\u56de\u306e\u554f\u984c\u306e\u5f62\u3092\u3057\u305f\u7d20\u6570\u304c\u7121\u9650\u306b\u3042\u308b\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u3066\u3044\u308b\u3068\u306f\u5230\u5e95\u601d\u3048\u307e\u305b\u3093. \u3088\u3063\u3066\u4eca\u56de\u306e\u3088\u3046\u306a, \u7279\u5b9a\u306e\u5f62\u3092\u3057\u305f\u7d20\u6570\u3092\u5217\u6319\u3059\u308b\u554f\u984c\u306f, \u6709\u9650\u500b\u3057\u304b\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u3092\u793a\u3059\u3053\u3068, \u304c\u5e38\u5957\u624b\u6bb5\u3068\u306a\u3063\u3066\u3044\u307e\u3059.<br><br>\u3067\u306f\u89e3\u7b54\u306b\u53c2\u308a\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n<p class=\"is-style-crease\">\\(p\\), \\(q\\)\u304c\u5171\u306b\u5947\u6570\u306e\u7d20\u6570\u306e\u3068\u304d, \\(p^q+q^p\\)\u306f\\(2\\)\u3088\u308a\u5927\u304d\u3044\u5076\u6570\u3068\u306a\u308b\u306e\u3067, \\(p\\), \\(q\\)\u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u306f\u5076\u6570\u306e\u7d20\u6570, \u3064\u307e\u308a\\(2\\)\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308b. \u3053\u3053\u3067\u306f, \\(q=2\\)\u3068\u3057\u3066\u3082\u4e00\u822c\u6027\u3092\u5931\u308f\u306a\u3044.<br><br>\\(q=2\\)\u306e\u3068\u304d, \\(p^2+2^p\\)\u304c\u7d20\u6570\u3068\u306a\u308b\u7d20\u6570\\(p\\)\u306b\u3064\u3044\u3066\u8003\u3048\u308b.<br><br>\\(p=2\\)\u3068\u3059\u308b\u3068, \\(p^2+2^p=2^2+2^2=8\\)\u3068\u306a\u308a, \u3053\u308c\u306f\u7d20\u6570\u3067\u306f\u306a\u3044.<br><br>\\(p=3\\)\u3068\u3059\u308b\u3068, \\(p^2+2^p=3^2+2^3=17\\)\u3068\u306a\u308a, \u3053\u308c\u306f\u7d20\u6570\u3067\u3042\u308b.<br><br>\u6b21\u306b, \\(p\\)\u3092\\(3\\)\u3088\u308a\u5927\u304d\u3044\u7d20\u6570\u3068\u3057, \\(p^2+2^p\\)\u304c\\(3\\)\u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b. \\(3\\)\u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u305f\u3044\u306e\u3067, \\(p^2\\), \\(2^p\\)\u3092\u305d\u308c\u305e\u308c\\(\\mbox{mod} 3\\)\u3067\u8003\u3048\u308b.<br><br>\u2460 \\(p^2\\)\u306b\u95a2\u3057\u3066<br>\\(p\\)\u306f\\(3\\)\u306e\u500d\u6570\u3067\u306a\u3044\u304b\u3089, \u81ea\u7136\u6570\\(n\\)\u3092\u7528\u3044\u3066, <br>$$p=3n\\pm 1$$<br>\u306e\u5f62\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.\u3053\u306e\u3068\u304d, <br>$$p^2=(3n\\pm 1)^2=9n^2\\pm 6n+1\\equiv 1 \\pmod 3$$<br>\u3068\u306a\u308b. <br><br>\u2461\\(2^p\\)\u306b\u95a2\u3057\u3066<br>\\(p\\)\u306f3\u3088\u308a\u5927\u304d\u3044\u7d20\u6570\u3088\u308a, \u5947\u6570\u3067\u3042\u308a, \\(2\\)\u4ee5\u4e0a\u306e\u81ea\u7136\u6570\\(m\\)\u3092\u7528\u3044\u3066\\(p=2m+1\\)\u3068\u8868\u305b\u308b. \u3053\u306e\u3068\u304d, <br>$$2^p=2^{2m+1}=2^{2m}\\times 2=4^m\\times 2 \\equiv 1^m\\times 2=2 \\pmod 3$$<br>\u3068\u306a\u308b.<br><br>\u2460, \u2461\u3088\u308a$$p^2+2^p\\equiv 1+2=3\\equiv 0 \\pmod 3$$\u3068\u306a\u308a, \\(p^2+2^p\\)\u306f\\(3\\)\u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u3053\u3067, \\(p\\)\u306f\\(3\\)\u3088\u308a\u5927\u304d\u3044\u7d20\u6570\u306a\u306e\u3067, \\(p^2+2^p\\)\u3082\\(3\\)\u3088\u308a\u5927\u304d\u3044\u3053\u3068\u304c\u308f\u304b\u308a, \\(p^2+2^p\\)\u304c\u7d20\u6570\u3068\u306a\u308b\u3053\u3068\u306f\u306a\u3044.<br><br>\u4ee5\u4e0a\u3088\u308a, \u7d20\u6570\\(p, q\\)\u3092\u7528\u3044\u3066\\(p^q+q^p\\)\u306e\u5f62\u3067\u8868\u305b\u308b\u7d20\u6570\u306f, <br>$$(p,q)=(2,3), (3,2)$$<br>\u306e\u3068\u304d\u306e\u307f\u3067\u3042\u308a, \u305d\u306e\u7d20\u6570\u306f\\(17\\)\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n<p>\u4eac\u90fd\u5927\u5b66\u306e\u554f\u984c\u3067\u3059\u304c, \u610f\u5916\u3068\u7c21\u5358\u3067\u3057\u305f\u306d. \u89e3\u7b54\u306e\u4e2d\u3067\u306f\u8003\u5bdf\u3092\u7701\u3044\u3066\u3044\u307e\u3059\u304c, \\(p^2+2^p\\)\u306e\u5f62\u3092\u3057\u305f\u7d20\u6570\u3092\u63a2\u3059\u969b\u306b, \u5b9f\u969b\u306b\\(p\\)\u306b\u5177\u4f53\u7684\u306a\u6570\u3092\u5165\u308c\u3066\u50be\u5411\u3092\u63b4\u3080\u3053\u3068\u304c\u5927\u4e8b\u3067\u3059. <br><br>\u5b9f\u969b, \\(p=5\\)\u306e\u3068\u304d, \\(p^2+2^p=5^2+2^5=57\\)\u3068\u306a\u308a, \u3053\u308c\u306f\\(3\\)\u3088\u308a\u5927\u304d\u3044\\(3\\)\u306e\u500d\u6570\u3068\u306a\u308a\u7d20\u6570\u3067\u306f\u3042\u308a\u307e\u305b\u3093. <br><br>\u307e\u305f, \\(p=7\\)\u306e\u3068\u304d, \\(p^2+2^p=7^2+2^7=177=3\\times 59\\)\u3068\u306a\u308a\u3053\u3061\u3089\u3082\\(3\\)\u306e\u500d\u6570\u3067\u3059. <br><br>\u3053\u306e\u8fba\u308a\u307e\u3067\u8a08\u7b97\u3057\u3066, \u300c\u3042\u3041\\(p\\)\u304c3\u3088\u308a\u5927\u304d\u3044\u7d20\u6570\u306e\u3068\u304d, \\(p^2+2^p\\)\u306f\\(3\\)\u306e\u500d\u6570\u306b\u306a\u308a\u305d\u3046\u3060\u306a\u300d\u3068\u6c17\u3065\u304f\u3053\u3068\u304c\u5927\u4e8b\u3067\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\/CNTZD9uMWXo?si=s1gReWUfwHsOq5eI\" 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\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059. \\(p\\), \\(q\\)\u3092\u7d20\u6570\u3068\u3059\u308b\u3068\u304d, \\(p^q+q^p\\)\u306e\u5f62\u3067\u8868\u305b\u308b\u7d20\u6570\u3092\u5168\u3066\u6c42\u3081\u3088. (2016 \u4eac\u90fd\u5927\u5b66) \u3053\u308c\u306f2016\u5e74\u306e\u4eac\u90fd\u5927\u5b66\u306e\u5165\u8a66\u554f\u984c\u3067\u51fa\u984c\u3055\u308c\u305f\u6709\u540d\u554f\u984c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":120,"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-52","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\/52","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=52"}],"version-history":[{"count":36,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/52\/revisions"}],"predecessor-version":[{"id":2023,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/52\/revisions\/2023"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/media\/120"}],"wp:attachment":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=52"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=52"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=52"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}