{"id":322,"date":"2025-07-04T20:44:15","date_gmt":"2025-07-04T11:44:15","guid":{"rendered":"https:\/\/math-friend.com\/?p=322"},"modified":"2025-07-29T10:35:40","modified_gmt":"2025-07-29T01:35:40","slug":"%e3%80%90%e4%b9%9d%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%918%e3%81%a7%e5%89%b2%e3%81%a3%e3%81%9f%e3%81%82%e3%81%be%e3%82%8a%e3%81%ab%e6%b3%a8%e7%9b%ae%e3%81%99%e3%82%8b%e6%95%b4%e6%95%b0","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=322","title":{"rendered":"\u3010\u4e5d\u5dde\u5927\u5b66\u5165\u8a66\u30118\u3067\u5272\u3063\u305f\u4f59\u308a\u306b\u6ce8\u76ee\u3059\u308b\u6574\u6570\u554f\u984c(2025)"},"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\">(1) \u6574\u6570\\(n\\) \u306b\u5bfe\u3057\u3066, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f, \\(0, 1, 4\\)\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<br>(2) \\(m\\), \\(n\\)\u3092\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d, \\(2^m=n^2+3\\)\u3092\u307f\u305f\u3059\\((m,n)\\)\u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<br><span style=\"text-align:right;display:block;\">(2025 \u4e5d\u5dde\u5927\u5b66\u7406\u7cfb[3])<\/span><\/p>\n\n\n\n<p>\u6574\u6570\u554f\u984c\u3092\u89e3\u304f\u969b\u306b\u300c\u4f59\u308a\u300d\u3092\u8003\u3048\u308b\u3053\u3068\u306f\u5e38\u5957\u624b\u6bb5\u3067\u3059. \u5b9f\u969b\u3053\u306e\u554f\u984c\u306f, \u6574\u6570\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306b\u6ce8\u76ee\u3057\u307e\u3059. \u3053\u306e\u3088\u3046\u306a\u554f\u984c\u3067\u306fmod\u3092\u4f7f\u3046\u3068\u8a18\u8ff0\u304c\u697d\u306a\u306e\u3067\u3059\u304c, \u9ad8\u6821\u3067\u7fd2\u3046\u3068\u306f\u9650\u3089\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3067\u3001mod\u3092\u4f7f\u308f\u305a\u611a\u76f4\u306b\u8a08\u7b97\u3092\u884c\u3046\u3053\u3068\u306b\u3057\u307e\u3059.\u305d\u308c\u3067\u306f\u554f\u984c\u3092\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) \u6574\u6570\\(n\\)\u306f\\(8\\)\u3067\u5272\u3063\u305f\u3042\u307e\u308a\u306b\u6ce8\u76ee\u3059\u308b\u3053\u3068\u3067, \u6574\u6570\\(k\\)\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e5\u30d1\u30bf\u30fc\u30f3\u306e\u3044\u305a\u308c\u304b\u306e\u5f62\u3067\u8868\u305b\u308b.<br>$$<br>n=8k, n=8k\\pm1, n=8k\\pm2, n=8k\\pm3, n=8k+4<br>$$<br>\u3053\u306e\u30d1\u30bf\u30fc\u30f3\u3054\u3068\u306b, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u3042\u307e\u308a\u3092\u8a08\u7b97\u3059\u308b.<br><br>\u2460 \\(n=8k\\)\u3068\u8868\u305b\u308b\u3068\u304d(\\(n\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\(0\\)\u306e\u3068\u304d)<br>\\(n^2=64k^2=8\\cdot 8k^2\\)\u3088\u308a, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(0\\)\u3067\u3042\u308b.<br><br>\u2461 \\(n=8k\\pm 1\\)\u3068\u8868\u305b\u308b\u3068\u304d(\u3064\u307e\u308a, \\(n\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\(1\\)\u307e\u305f\u306f\\(7\\)\u306e\u3068\u304d)<br>\\(n^2=(8k\\pm 1 )^2=64k^2\\pm 16k+1=8(8k^2\\pm 2k)+1\\)\u3088\u308a, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(1\\)\u3067\u3042\u308b.<br><br>\u2462 \\(n=8k\\pm 2\\)\u3068\u8868\u305b\u308b\u3068\u304d(\u3064\u307e\u308a, \\(n\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\(2\\)\u307e\u305f\u306f\\(6\\)\u306e\u3068\u304d)<br>\\(n^2=(8k\\pm 2 )^2=64k^2\\pm 32k+4=8(8k^2\\pm 4k)+4\\)\u3088\u308a, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(4\\)\u3067\u3042\u308b.<br><br>\u2463 \\(n=8k\\pm 3\\)\u3068\u8868\u305b\u308b\u3068\u304d(\u3064\u307e\u308a, \\(n\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\(3\\)\u307e\u305f\u306f\\(5\\)\u306e\u3068\u304d)<br>\\(n^2=(8k\\pm 3 )^2=64k^2\\pm 48k+9=8(8k^2\\pm 6k+1)+1\\)\u3088\u308a, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(1\\)\u3067\u3042\u308b.<br><br>\u2464 \\(n=8k+4\\)\u3068\u8868\u305b\u308b\u3068\u304d(\u3064\u307e\u308a, \\(n\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\(4\\)\u306e\u3068\u304d)<br>\\(n^2=(8k+ 4 )^2=64k^2+ 64k+16=8(8k^2+ 8k+2)\\)\u3088\u308a, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(0\\)\u3067\u3042\u308b.<br><br>\u4ee5\u4e0a\u304b\u3089, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f, \\(0, 1, 4\\)\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b.<\/p>\n\n\n\n<p>(2) \\(2^m=n^2+3\\)\u306e\u53f3\u8fba\\(n^2+3\\)\u306f\u3001(1)\u3088\u308a\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f, \\(3(=0+3), 4(=1+3), 7(=4+3)\\)\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308a, \u3053\u308c\u304c\\(8\\)\u306e\u500d\u6570\u306b\u306a\u308b\u3053\u3068\u306f\u306a\u3044. \u4e00\u65b9, \u5de6\u8fba\u306e\\(2^m\\)\u306f\\(m\\geq 3\\)\u306e\u3068\u304d\\(8\\)\u306e\u500d\u6570\u306b\u306a\u308b\u304b\u3089, \u7b49\u5f0f\u3092\u6e80\u305f\u3059\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u306e\u7d44\\((m,n)\\)\u304c\u5b58\u5728\u3059\u308b\u3068\u3059\u308c\u3070, \\(m\\)\u306f\\(0, 1, 2\\)\u306e\u3044\u305a\u308c\u304b\u3068\u306a\u308b.<br><br>\u2460 \\(m=0\\)\u306e\u3068\u304d<br>\u5de6\u8fba\u306f\\(1\\)\u3068\u306a\u308a, \u53f3\u8fba\\(n^2+3\\)\u306f\\(3\\)\u4ee5\u4e0a\u306e\u305f\u3081, \u3053\u308c\u3092\u6e80\u305f\u3059\\(n\\)\u306f\u5b58\u5728\u3057\u306a\u3044.<br><br>\u2461 \\(m=1\\)\u306e\u3068\u304d<br>\u5de6\u8fba\u306f\\(2\\)\u3068\u306a\u308a, \u53f3\u8fba\\(n^2+3\\)\u306f\\(3\\)\u4ee5\u4e0a\u306e\u305f\u3081, \u3053\u308c\u3092\u6e80\u305f\u3059\\(n\\)\u306f\u5b58\u5728\u3057\u306a\u3044.<br><br>\u2462 \\(m=2\\)\u306e\u3068\u304d<br>\u5de6\u8fba\u306f\\(4\\)\u3067\u3042\u308b\u304b\u3089, <br>$$<br>4=n^2+3\\iff n^2=1 \\iff n=\\pm 1<br>$$<br>\u3068\u306a\u308b. \u3053\u3053\u3067\\(n\\)\u306f\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u3060\u304b\u3089, \\(n=1\\)\u306e\u307f\u304c\u89e3\u3068\u306a\u308b. \u3088\u3063\u3066\u7b49\u5f0f\u3092\u6e80\u305f\u3059\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u306e\u7d44\u306f\\((m,n)=(2,1)\\)\u3067\u3042\u308a, \u3053\u308c\u304c\u5168\u3066\u3067\u3042\u308b.<\/p>\n<\/div><\/div>\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\/01VmIP4rnlQ?si=yGBnp_IqO4RFCbgi\" 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. (1) \u6574\u6570\\(n\\) \u306b\u5bfe\u3057\u3066, \\(n^2\\)\u3092\\(8\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f, \\(0, 1, 4\\)\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.(2) \\(m\\), \\(n\\)\u3092\\(0\\)\u4ee5\u4e0a\u306e\u6574 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":354,"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-322","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\/322","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=322"}],"version-history":[{"count":7,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/322\/revisions"}],"predecessor-version":[{"id":2036,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/322\/revisions\/2036"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/media\/354"}],"wp:attachment":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=322"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=322"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=322"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}