{"id":14660,"date":"2023-06-09T09:32:09","date_gmt":"2023-06-09T00:32:09","guid":{"rendered":"https:\/\/zelkova-tree.net\/WordPress\/?p=14660"},"modified":"2023-06-09T09:41:05","modified_gmt":"2023-06-09T00:41:05","slug":"an-efficient-factoring-algorithm-by-repunit-number-method-%e3%82%92%e8%aa%ad%e3%82%80","status":"publish","type":"post","link":"https:\/\/zelkova-tree.net\/WordPress\/2023\/06\/09\/an-efficient-factoring-algorithm-by-repunit-number-method-%e3%82%92%e8%aa%ad%e3%82%80\/","title":{"rendered":"An Efficient Factoring Algorithm by Repunit Number Method \u3092\u8aad\u3080"},"content":{"rendered":"<p><font face=\"\u30e1\u30a4\u30ea\u30aa\">\u3069\u3046\u3082\uff0c\u306a\u304b\u306a\u304b\u8a71\u304c\u5408\u308f\u306a\u3044\uff0eEFM\u3067\u306f[8\u2019]\u3068\u3057\u3066\uff0c<\/p>\n<p>Let p be a prime number which is neither 2 nor 5, and the recursion unit of p be U, then pU\/9 is a repunit number Rn, i.e., pU=10n-1.<\/p>\n<p>\u3068\u3044\u3046\u547d\u984c\u3092\u63d0\u793a\u3057\uff0c\u305d\u306e\u8a3c\u660e\u3092\u4e0e\u3048\u3066\u3044\u308b\u304c\uff0c\u305d\u306e\u4e2d\u3067\uff50\u304c\u7d20\u6570\u3067\uff50\u3068e\u304c\u4e92\u3044\u306b\u7d20\u306e\u3068\u304d\uff0c\uff42^e-1\u22610 mod p \u3068\u306a\u308b\u3088\u3046\u306a\u6700\u5c0f\u306e\u6574\u6570e\u304c\u5b58\u5728\u3057\uff0c\u3053\u308c\u306f\uff50\u304cb^n-1\u3092\u5272\u308a\u5207\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3068\u3057\u3066\u3044\u308b\uff0e\u3053\u308c\u306f\uff0c\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u5c0f\u5b9a\u7406\u3092\u8a00\u3044\u63db\u3048\u305f\u3060\u3051\u306e\u3082\u306e\u3067\u3042\u308b\u304b\u3089\uff0c\u6b63\u3057\u3044\uff0e\u5f93\u3063\u3066\uff0cpU=b^e-1\u3068\u306a\u308b\u3088\u3046\u306a\u6700\u5c0f\u306e\u6574\u6570U\u304c\u5b58\u5728\u3059\u308b\u3068\u3057\uff0c\u3053\u308c\u306fU\u304cp\u306e\u5faa\u74b0\u5358\u4f4d\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3068\u3057\u3066\u3044\u308b\uff0e<\/p>\n<p>n=11, e=2, K=7, B=5\u306e\u3068\u304d\uff0c\uff20\uff1d\uff15, U=284, nU=4444\uff0crep=1111\u3067\uff0cU\u306f11\u306e\u5faa\u74b0\u5358\u4f4d\u3067\u3042\u308b\uff0e\u3053\u306e\u3068\u304d\uff0c5^5%11=284\u20261\u3068\u306a\u308b\u306e\u3067\uff0cB^@%n=U\u20261\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\uff0e\u3053\u3053\u3067\uff0c\uff20\u306f\u5faa\u74b0\u7bc0\u306e\u6841\u6570\u3067\u3042\u308b\uff0ee\u3068\u3044\u3046\u6307\u6570\u306f\u3079\u304d\u5270\u4f59\u6570\u5217\u306e\u03c8\u306b\u76f8\u5f53\u3059\u308b\u6570\u3068\u8003\u3048\u3089\u308c\u308b\u304c\uff0cQ=264\u306b\u76f8\u5f53\u3059\u308b\u6570\u304c\u03c8\u306a\u3044\u3057#\u306b\u51fa\u3066\u304f\u308b\u3088\u3046\u306a\u8a08\u7b97\u5f0f\u3092\u7acb\u3066\u3089\u308c\u308b\u3060\u308d\u3046\u304b\uff1f#\u306a\u3044\u3057\u03c8\u306fK\u3088\u308a\u5c0f\u3055\u3044\u3068\u8003\u3048\u3089\u308c\u308b\u306e\u3067\uff0c\u76f8\u5f53\u5927\u304d\u306aK\u304c\u5fc5\u8981\u306b\u306a\u308b\uff0en\u3082\u304b\u306a\u308a\u5927\u304d\u306a\u6570\u3067\u306a\u304f\u3066\u306f\u306a\u3089\u306a\u3044\u3060\u308d\u3046\uff0eK = 1111\u306b\u8a2d\u5b9a\u3059\u308b\u3068\u304b\u306a\u308a\u8fd1\u3044\u6570\u5b57\u306b\u8fd1\u3065\u304f\uff0eK\u304c\u7d20\u6570\u3067\u306a\u3044\u3068\u5927\u304d\u306a\u6570\u304c\u51fa\u3066\u3053\u306a\u3044\uff0e\u3061\u3087\u3063\u3068\u7684\u304c\u5916\u308c\u3066\u3044\u308b\u611f\u3058\uff0e<\/p>\n<p>N^e\u306e\u8a08\u7b97\u5024\u3092\u518d\u5229\u7528\u3059\u308b\u305f\u3081\u306b\u51fa\u529b\u7528\u306e\u30c6\u30ad\u30b9\u30c8\u30dc\u30c3\u30af\u30b9\u3092\u8ffd\u52a0\u3057\u3088\u3046\u3068\u601d\u3063\u305f\u306e\u3060\u304c\uff0c\u3059\u3067\u306b\u5341\u4e8c\u5206\u306b\u753b\u9762\u304c\u6df7\u307f\u5408\u3063\u3066\u3044\u308b\u306e\u3067\u51fa\u529b\u30da\u30fc\u30f3\u306b\u66f8\u304d\u51fa\u3059\u3053\u3068\u306b\u3059\u308b\uff0eEFA\u3068\u5b9f\u88c5\u304c\u5408\u308f\u306a\u3044\u70b9\u3068\u3057\u3066\uff0c09 Mar 2005\u306e\u30e1\u30fc\u30eb\u306b\u306f For example 21 is a composite but has a repunit number 111 with recursion unit 03 \u3068\u3042\u308b\u304c\uff0c\u3053\u308c\u306f\u9053\u5177\u7bb1\u306e\u52d5\u4f5c\u3068\u3042\u3063\u3066\u3044\u306a\u3044\uff0e1\/21=0.04761904761904761904761904\u2026\u3067\uff0cU=047619\uff0cnU=999999, rep=111111\u3060\uff0e\u3053\u308c\u306f\u660e\u3089\u304b\u306b\u8aa4\u8a8d\u3068\u601d\u308f\u308c\u308b\uff0e\u3053\u306e\u30e1\u30fc\u30eb\u306b\u8a18\u8f09\u3055\u308c\u305f\u30ea\u30b9\u30c8\u3067\u306f\uff0cU=047619\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u4f55\u304b\u52d8\u9055\u3044\u3057\u3066\u3044\u305f\u306e\u3067\u306f\u306a\u3044\u3060\u308d\u3046\u304b\uff1f<\/p>\n<p>EFA\u3067\u306fn\u2192r{m}(\uff4d\u6841\u306e\u30ec\u30d4\u30e5\u30cb\u30c3\u30c8\u6570)\u306e\u3068\u304d\uff0cn\u3092r{m}\u306e\u30aa\u30ea\u30b8\u30cd\u30fc\u30bf\u3068\u547c\u3093\u3067\u3044\u308b\uff0e\u3053\u306e\u3068\u304d\uff0c\uff4e\u306e\u7d20\u56e0\u6570\u5206\u89e3\u3068r{m}\u306b\u306f\u660e\u3089\u304b\u306b\u4f55\u3089\u304b\u306e\u76f8\u95a2\u304c\u3042\u308b\uff0e\uff4e\u306f\u5fc5\u305a\u3057\u3082\u30ec\u30d4\u30e5\u30cb\u30c3\u30c8\u6570\u306e\u30aa\u30ea\u30b8\u30cd\u30fc\u30bf\u306b\u306a\u308b\u8a33\u3067\u306f\u306a\u3044\u306e\u3067\uff0c\uff4e\u304b\u3089\u751f\u6210\u3055\u308c\u308b\u30ec\u30d4\u30e5\u30cb\u30c3\u30c8\u985e\u4f3c\u306e\u6570\u3092rep(n)\u3068\u8868\u8a18\u3059\u308b\u3053\u3068\u306b\u3057\u3088\u3046\uff0erep(n)\u304c\u30ec\u30d4\u30e5\u30cb\u30c3\u30c8\u6570r(m)\u306b\u306a\u308b\u6761\u4ef6\u3092\u8abf\u3079\u308b\u3053\u3068\u306f\u610f\u5473\u304c\u3042\u308b\u3088\u3046\u306b\u601d\u308f\u308c\u308b\uff0e<\/p>\n<p><font color=\"#cccccc\">\u25b2<\/font>B\u306e\u7d20\u56e0\u6570\u5206\u89e3\u3082\u51fa\u3059\u5fc5\u8981\u304c\u3042\u308b\uff0e\u30ec\u30d4\u30e5\u30cb\u30c3\u30c8\u306b\u95a2\u308f\u308a\u304c\u3042\u308b\uff0e<\/p>\n<p>09 Mar 2005\u306e\u30e1\u30fc\u30eb\u306b\u306f\u8aa4\u308a\u304c\u3042\u308b\u304c\uff0c\u305d\u306e\u4e0b\u306e\u65b9\u306e\u30ea\u30b9\u30c8\u306f\u6b63\u3057\u3044\u306e\u3067\uff0c\u6ce8\u610f\u6df1\u3044\u8aad\u8005\u306a\u3089\u591a\u5206\u6c17\u4ed8\u3044\u305f\u3053\u3068\u3060\u308d\u3046\uff0e\u3053\u306e\u30e1\u30fc\u30eb\u306e\u7d42\u308f\u308a\u306e\u65b9\u306b\uff0c\uff50\u304c\u7d20\u6570\u3067\u3042\u308b\u305f\u3081\u306e\uff15\u6761\u4ef6\u3068\u3044\u3046\u306e\u3092\u5217\u6319\u3057\u3066\u3044\u308b\uff0e\u3053\u306e\u9805\u76ee\u306f\u30c1\u30a7\u30c3\u30af\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\uff0e\u3082\u3057\uff0c\u3053\u308c\u3089\u304c\u6b63\u3057\u3044\u3068\u3059\u308b\u3068\u7d20\u6570\u5224\u5b9a\u3092\u304b\u306a\u308a\/\u3042\u308b\u7a0b\u5ea6\u9ad8\u901f\u5316\u3067\u304d\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\uff0e<\/p>\n<p>(1) p has a repunit R_k=pU(p)\/9<br \/>(2) k divides p-1<br \/>(3) for p &gt; 3, (p,9)=1<br \/>(4) p divides R_k<br \/>(5) p and k are mutually prime<\/p>\n<p><\/p>\n<p><\/p>\n<p><\/font><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u3069\u3046\u3082\uff0c\u306a\u304b\u306a\u304b\u8a71\u304c\u5408\u308f\u306a\u3044\uff0eEFM\u3067\u306f[8\u2019]\u3068\u3057\u3066\uff0c Let p be a prime number which is neither 2 nor 5, and the recursion unit of p be  &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/zelkova-tree.net\/WordPress\/2023\/06\/09\/an-efficient-factoring-algorithm-by-repunit-number-method-%e3%82%92%e8%aa%ad%e3%82%80\/\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;An Efficient Factoring Algorithm by Repunit Number Method \u3092\u8aad\u3080&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/posts\/14660"}],"collection":[{"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/comments?post=14660"}],"version-history":[{"count":10,"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/posts\/14660\/revisions"}],"predecessor-version":[{"id":14670,"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/posts\/14660\/revisions\/14670"}],"wp:attachment":[{"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/media?parent=14660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/categories?post=14660"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/zelkova-tree.net\/WordPress\/wp-json\/wp\/v2\/tags?post=14660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}