\u7d20\u6570\u7684\u5b9a\u4e49\uff1a\u5982\u679c\u4e00\u4e2a\u5927\u4e8e1\u7684\u6b63\u6574\u6570n\u53ea\u67091\u548c\u5b83\u672c\u8eab\u8fd9\u4e24\u4e2a\u56e0\u5b50\uff0c\u90a3\u4e48n\u662f\u7d20\u6570\uff0c\u5426\u5219n\u662f\u5408\u6570\u30021\u4e0d\u662f\u7d20\u6570\u4e5f\u4e0d\u662f\u5408\u6570\u3002<\/p>\n
\u6c42\u7d20\u6570\u7684\u51e0\u79cd\u65b9\u6cd5\uff0c\u590d\u6742\u5ea6\u4ece\u5927\u5230\u5c0f\uff1a<\/p>\n
\u5bf9\u4e8e\u4e00\u4e2a\u6574\u6570n\uff0c\u5982\u679c\u533a\u95f4 \u6b32\u5224\u65ad \u7ebf\u6027\u5904\u7406\u7d20\u6570\u7684\u65b9\u6cd5\uff0c\u5229\u7528\u552f\u4e00\u5206\u89e3\u5b9a\u7406\u5f97\u5230\u4e00\u4e2a\u6027\u8d28\uff1a\u6bcf\u4e2a\u5408\u6570\u603b\u4f1a\u6709\u4e00\u4e2a\u6700\u5c0f\u7684\u8d28\u56e0\u6570\uff0c\u5229\u7528\u8fd9\u4e2a\u6700\u5c0f\u7684\u8d28\u56e0\u6570\u6765\u6807\u8bb0\u8be5\u5408\u6570\uff0c\u6bcf\u4e2a\u6570\u53ea\u4f1a\u88ab\u6807\u8bb01\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \u7c73\u52d2\u7f57\u5bbe\u7d20\u6027\u6d4b\u8bd5\u53ef\u4ee5\u5728\u8f83\u77ed\u7684\u65f6\u95f4\u5185\u5224\u65ad\u51fa\u4e00\u4e2a\u5927\u6574\u6570\u7684\u7d20\u6027\u3002\u6bd4\u5982\u73b0\u5728\u60f3\u5224\u65ad\u4e00\u4e2a\u957f\u5ea6100\u4f4d\u7684\u6574\u6570\u662f\u5426\u4e3a\u7d20\u6570\uff0c\u4f7f\u7528\u4e0a\u9762\u7684\u51e0\u79cd\u65b9\u6cd5\u90fd\u4e0d\u80fd\u5728\u77ed\u65f6\u95f4\u5185\u5f97\u5230\u7ed3\u679c\uff0c\u56e0\u6b64\u5f88\u96be\u5f97\u5230\u4e00\u4e2a\u786e\u5b9a\u7684\u7ed3\u679c\u6765\u5224\u65ad\u8be5\u6574\u6570\u662f\u5426\u4e3a\u7d20\u6570\u3002\u53ef\u4ee5\u8003\u8651\u5bf9\u6574\u6570\u8fdb\u884c\u7d20\u6027\u6d4b\u8bd5\uff0c\u5177\u4f53\u9700\u8981\u7528\u5230\u4e0b\u9762\u4e24\u4e2a\u5b9a\u7406\u3002 \u4e8c\u6b21\u63a2\u6d4b\u5b9a\u7406<\/strong>\uff1a\u82e5n\u4e3a\u8d28\u6570\uff0c\u5219\u65b9\u7a0b \u7c73\u52d2\u7f57\u5bbe\u7d20\u6027\u6d4b\u8bd5\u6b65\u9aa4\u4e3a\uff1a\u6bcf\u4e00\u8f6e\u6d4b\u8bd5\u4e2d\uff0c\u9009\u53d6\u4e00\u4e2a\u6d4b\u8bd5\u56e0\u5b50a\uff0c\u5148\u8ba1\u7b97 \u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6b65\u9aa4\u548c\u4e0a\u8ff0\u662f\u53cd\u7740\u6765\u7684\uff1a\u5148\u5bf9n-1\u4e0d\u65ad\u5f00\u65b9\uff0c\u5373\u8868\u793a\u6210 Java\u4ee3\u7801\uff0cC++\u5c31\u5148\u6682\u65f6\u653e\u7740\u5427...\u6709\u65f6\u95f4\u6574\u7406\u4e00\u4e0b\u5927\u6574\u6570\u518d\u5199<\/p>\n \u4ee3\u7801\u4e2d\u6211\u9009\u53d6\u7684\u6d4b\u8bd5\u56e0\u5b50a\u662f\u4ece16350\u5f00\u59cb\u7684\u8fde\u7eed\u51e0\u4e2a\u6570\uff0c\u9009\u4e86\u51e0\u4e2a\u6570\u5c31\u662f\u8fdb\u884c\u51e0\u8f6e\u6d4b\u8bd5\uff0c\u6d4b\u8bd5\u51fa\u9519\u7684\u6539\u7387\u4e3a \u7d20\u6570\u7684\u5b9a\u4e49\uff1a\u5982\u679c\u4e00\u4e2a\u5927\u4e8e1\u7684\u6b63\u6574\u6570n\u53ea\u67091\u548c\u5b83\u672c\u8eab\u8fd9\u4e24\u4e2a\u56e0\u5b50\uff0c\u90a3\u4e48n\u662f\u7d20\u6570\uff0c\u5426\u5219n\u662f\u5408\u6570\u30021\u4e0d\u662f\u7d20\u6570 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[60],"tags":[],"_links":{"self":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts\/1106"}],"collection":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/comments?post=1106"}],"version-history":[{"count":9,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts\/1106\/revisions"}],"predecessor-version":[{"id":1124,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts\/1106\/revisions\/1124"}],"wp:attachment":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/media?parent=1106"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/categories?post=1106"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/tags?post=1106"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}[2, \\sqrt(n)]<\/code>\u5185\u4e0d\u5b58\u5728n\u7684\u56e0\u5b50\uff0c\u90a3\u4e48n\u662f\u8d28\u6570\u3002\u56e0\u4e3a\u4e00\u4e2a\u6570\u7684\u56e0\u5b50\u4e2d\u9664\u4e86\u5176\u7b97\u672f\u5e73\u65b9\u6839\u5916\uff0c\u90fd\u662f\u6210\u5bf9\u51fa\u73b0\u7684\uff0c\u4e14\u4e0d\u53ef\u80fd\u5b58\u5728\u4e24\u4e2a\u5927\u4e8e\\sqrt(n)<\/code>\u7684\u56e0\u5b50(\u663e\u800c\u6613\u89c1)\u3002\u90a3\u4e48\u4f7f\u7528\u8bd5\u9664\u6cd5\u5224\u65ad\u4e00\u4e2a\u7d20\u6570\u6700\u591a\u9700\u8981O(\\sqrt(n))<\/code>\u7684\u590d\u6742\u5ea6\uff0c\u5224\u65adn\u4e2a\u5c31\u662fO(n\\sqrt{n})<\/code><\/p>\n\u57c3\u5f0f\u7b5b\u6cd5<\/h5>\n
[2, n]<\/code>\u5185\u7684\u6570\u7684\u7d20\u6027\uff0c\u4ece\u524d\u5f80\u540e\u4e00\u4e2a\u4e00\u4e2a\u626b\u63cf\uff0c\u82e5\u5f53\u524d\u6570\u672a\u505a\u5408\u6570\u6807\u8bb0\uff0c\u90a3\u4e48\u5b83\u5c31\u662f\u8d28\u6570\uff0c\u6bd4\u5982\u4e00\u5f00\u59cb\u5c31\u80fd\u5224\u65ad2\u548c3\u90fd\u662f\u8d28\u6570\uff1b\u6bcf\u626b\u63cf\u5230\u4e00\u4e2a\u8d28\u6570p\uff0c\u90a3\u4e48\u5b83\u7684\u4efb\u4f55\u4e0d\u5927\u4e8en\u7684\u500d\u6570\u90fd\u662f\u5408\u6570\uff0c\u5c31\u628a2p, 3p, 4p,,,\u4e00\u76f4\u5230n\u4e3a\u6b62\u8fd9\u4e9b\u6570\u90fd\u6807\u8bb0\u4e3a\u8d28\u6570\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u7565\u5927\u4e8eO(n)<\/code>\u3002
\n\u56e0\u4e3a\u5728\u6807\u8bb0\u7684\u65f6\u5019\u4ec5\u662f\u5728\u626b\u63cf\u5230\u8d28\u6570\u65f6\u624d\u5f00\u59cb\u6807\u8bb0\u5de5\u4f5c\uff0c\u82e5\u5bf9\u4e8e[1, n]<\/code>\u5185\u6bcf\u4e00\u4e2a\u6570\u90fd\u8fdb\u884c\u5982\u4e0a\u6807\u8bb0\uff0c\u90a3\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(nlogn)<\/code>\u3002\u5373\\sum_{i=1}^{n}n\/i \\approx nlogn<\/code>.<\/p>\n\u6b27\u62c9\u7b5b<\/h5>\n
O(n)<\/code>\u3002
\n\u6b65\u9aa4\uff1a\u540c\u6837\u662f\u4ece\u524d\u5f80\u540e\u4e00\u4e2a\u4e2a\u626b\u63cf\uff0c\u82e5\u626b\u63cf\u5230\u7684\u6570\u672a\u88ab\u6807\u8bb0\uff0c\u5b83\u5c31\u662f\u8d28\u6570\uff0c\u5e76\u5c06\u8be5\u8d28\u6570p\u5b58\u8fdb\u8d28\u6570\u6570\u7ec4prime\u91cc\uff0c\u7528cnt\u6765\u8bb0\u5f55\u6b64\u65f6\u5df2\u7ecf\u5f97\u5230\u591a\u5c11\u4e2a\u7d20\u6570\uff0cprime\u6570\u7ec4\u91cc\u7684\u8d28\u6570\u662f\u4ece\u5c0f\u5230\u5927\u6392\u5217\u597d\u7684\u3002\u518d\u626b\u63cfprime\u6570\u7ec4\uff0c\u5bf9\u4e8e\u90a3\u4e9b\u5c0f\u4e8ep\u7684<\/p>\nMiller-Rabin\u7d20\u6027\u6d4b\u8bd5<\/h5>\n
\n\u8d39\u9a6c\u5c0f\u5b9a\u7406<\/strong>\uff1a\u82e5n\u4e3a\u8d28\u6570\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u6574\u6570a\uff0c\u6709a^{n-1}(\\%n)=1<\/code>\uff0c\u4f46\u662f\u5f53\u8be5\u5f0f\u6210\u7acb\u65f6\uff0cn\u4e0d\u4e00\u5b9a\u662f\u8d28\u6570\uff1b\u5f53a^{n-1}(\\%n)<\/code>\u7684\u7ed3\u679c\u4e0d\u4e3a1\u662f\uff0cn\u4e00\u5b9a\u662f\u5408\u6570\u3002\u53ef\u4ee5\u4ee5\u6b64\u6765\u5728\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u6d4b\u8bd5\u4e00\u4e2a\u6570\u7684\u7d20\u6027\u3002(\u8d39\u9a6c\u5c0f\u5b9a\u7406\u63a8\u5e7f\u81ea\u6b27\u62c9\u5b9a\u7406<\/a>)<\/p>\nx^2(\\%n)=1<\/code>\u7684\u89e3\u53ea\u67091\u548cn-1.\u5bf9\u65b9\u7a0b\u8fdb\u884c\u53d8\u5f62\uff1ax^2-1=an, (x+1)(x-1)=an<\/code>\uff0cn\u4e3a\u7d20\u6570\u6240\u4ee5\u6ca1\u6709\u5176\u4f59\u7684\u8d28\u56e0\u5b50\uff0c\u65b9\u7a0b\u6210\u7acb\u7684\u60c5\u51b5\u53ea\u80fd\u662fx=1\u6216\u8005x=n-1\u3002\u8fd9\u91cc\u7684x\u90fd\u662f\u7ecf\u8fc7\u5bf9n\u53d6\u6a21\u540e\u7684\u503c\u3002<\/p>\na^{n-1}\\%n<\/code>\u7684\u503c\u662f\u5426\u4e3a1\uff0c\u4e0d\u662f1\u7684\u8bdd\u6d4b\u8bd5\u4e0d\u901a\u8fc7\uff0cn\u4e00\u5b9a\u4e3a\u5408\u6570\uff1b\u5426\u5219\u7ee7\u7eed\u4f7f\u7528\u4e8c\u6b21\u63a2\u6d4b\u5b9a\u7406\u8fdb\u884c\u6d4b\u8bd5\uff0c\u5bf9n-1\u8fdb\u884c\u5f00\u65b9\uff0c\u82e5\u5f00\u65b9\u5f97\u5230\u7684\u503c\u4e3an-1\uff0c\u6d4b\u8bd5\u901a\u8fc7\uff0cn\u53ef\u80fd\u4e3a\u8d28\u6570\uff1b\u82e5\u5f00\u65b9\u5f97\u5230\u7684\u503c\u4e3a1\uff0c\u90a3\u4e48\u53ef\u4ee5\u7ee7\u7eed\u5f00\u65b9\uff1b\u82e5\u5f97\u5230\u4e86\u5176\u5b83\u503c\uff0c\u90a3\u4e48\u6d4b\u8bd5\u4e0d\u901a\u8fc7\uff0cn\u4e00\u5b9a\u662f\u5408\u6570\u3002<\/p>\nn-1=2^{s}\u00b7d<\/code>\u7684\u5f62\u5f0f\uff0cd\u4e3a\u5947\u6570\uff0cs\u5927\u4e8e1\u3002
\n\u7136\u540e\u5148\u8ba1\u7b972^d\\%n<\/code>\u662f\u5426\u4e3a1\u6216n-1\uff0c\u82e5\u662f\uff0c\u5219\u6d4b\u8bd5\u901a\u8fc7\uff1b\u5426\u5219\uff0c\u4e00\u5b9a\u662f\u5728\u5f00\u65b9\u8fc7\u7a0b\u4e2d\u6709\u67d0\u4e00\u6b65\u6a21n\u51fa\u73b0\u4e86\u5176\u5b83\u503c\uff0c\u5c31\u5f00\u59cb\u5bf9\u8be5\u503c\u8fdb\u884c\u5e73\u65b9\u53d6\u6a21\u64cd\u4f5c\u3002\u82e5\u5148\u5f97\u5230n-1\uff0c\u5219\u6d4b\u8bd5\u901a\u8fc7\uff1b\u82e5\u5148\u5f97\u52301\uff0c\u5219\u8bf4\u660e\u5b58\u5728x\u4f7f\u5f97x^2=1<\/code>\u7684\u89e3\u4e0d\u662f1\u4e5f\u4e0d\u662fn-1\uff0c\u90a3\u4e48\u6d4b\u8bd5\u4e0d\u901a\u8fc7\uff0cn\u4e00\u5b9a\u4e3a\u5408\u6570\u3002\u82e5\u5df2\u7ecf\u5e73\u65b9\u8fc7s\u6b21\u540e\u8fd8\u672a\u9000\u51fa\u4e0a\u8ff0\u5faa\u73af\uff0c\u4e5f\u5c31\u662f\u8bf4\u6b64\u65f6\u8ba1\u7b97\u7684\u503c\u5df2\u7ecf\u8fbe\u5230\u4e86a^{n-1}<\/code>\uff0c\u90a3\u4e48\u6b64\u65f6\u4e0d\u8bba\u7ed3\u679c\u662f1\u8fd8\u662fn-1\uff0cn\u90fd\u4e00\u5b9a\u662f\u5408\u6570\u3002
\n\u56e0\u4e3a\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u8981\u7528\u5230\u9ad8\u7cbe\u5ea6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO((lgn)^3)<\/code>
\npython\u4ee3\u7801\u5982\u4e0b:<\/p>\ndef pmod(a, b, c):\n if (a==0):\n return 0\n ans = 1\n while b>0:\n if (b&1):\n ans = ans*a%c\n b >>= 1\n a = a*a%c\n return ans\n\ndef check(a, n):\n d = n-1\n s = 0\n while (d&1)==0:\n d >>= 1\n s+=1\n t = pmod(a, d, n)\n\n if t==1 or t==n-1:\n return True\n\n while (s>0):\n s -= 1\n\n if (t==n-1):\n return True\n if t==1:\n return False\n t = t*t%n\n\n return False\n\ndef solve(n):\n if (n<1000):\n f = 0\n for i in range(2, n):\n if n%i==0:\n f = 1\n break\n if f==0:\n print("Yes")\n else:\n print("No")\n return 0\n\n for a in range(16350, 16351):\n if False==check(a, n):\n print("No")\n return 0\n print("Yes")\n return 0\n\nwhile True:\n try:\n s = input()\n solve(int(s))\n except EOFError:\n break<\/code><\/pre>\nimport java.math.BigInteger;\nimport java.util.Scanner;\npublic class primeJudge{\n public static void main(String[] args) {\n Scanner scanner = new Scanner(System.in);\n while (scanner.hasNextLine()) {\n if ( false==isprime.primejudge(new BigInteger(scanner.nextLine())) ) System.out.println("No");\n else System.out.println("Yes");\n }\n scanner.close();\n }\n};\n\nclass isprime {\n\n public static boolean primejudge(BigInteger n) {\n if (n.subtract(new BigInteger("1000000")).signum() <= 0 ) {\n int tmp = Integer.valueOf(n.toString());\n for (int i=2; i*i<=tmp; i++) if (tmp%i==0)\n return false;\n return true;\n }\n for (int a = 16350; a<=16360; a++) {\n if (Miller_Rabin(n, new BigInteger(String.valueOf(a)))==false)\n return false;\n }\n return true;\n }\n \/\/ \u5feb\u901f\u5e42 a^b%m\n private static BigInteger qpow(BigInteger a, BigInteger b, BigInteger m, BigInteger yi, BigInteger er) {\n if (b.signum()==0) return new BigInteger("0");\n BigInteger ans = new BigInteger("1");\n while (b.signum()==1) {\n if (b.mod(er).equals(yi)) ans = (ans.multiply(a)).mod(m);\n b = b.divide(er);\n a = (a.multiply(a)).mod(m);\n }\n return ans;\n }\n \/\/ \u4e00\u6b21\u7c73\u52d2\u7f57\u5bbe\u6d4b\u8bd5\n private static boolean Miller_Rabin(BigInteger n, BigInteger a) {\n BigInteger d = new BigInteger("0");\n BigInteger yi = new BigInteger("1");\n BigInteger er = new BigInteger("2");\n BigInteger tmp = n.subtract(yi);\n d = n.subtract(yi);\n int s = 0;\n \/\/233System.out.println(d.mod(er).toString());\n while (d.mod(er).equals(yi)==false) {\n s++;\n d = d.divide(er);\n }\n BigInteger t = qpow(a, d, n, yi, er);\n if (t.equals(yi) || t.equals(tmp)) return true;\n while (s>0) {\n s -= 1;\n if (t.equals(tmp)) return true;\n if (t.equals(yi)) return false;\n t = t.multiply(t).mod(n);\n }\n return false;\n }\n};\n\/\/ \u5927\u698220\u6b21\u7c73\u52d2\u7f57\u5bbe\u6d4b\u8bd5\u6d88\u80171000ms\n<\/code><\/pre>\n\\frac{1}{2^{k}}<\/code>\uff0ck\u4e3a\u6d4b\u8bd5\u6b21\u6570\u3002
\n\u6211\u778e\u9009\u7684\u8fd9\u4e2a\u6570\u51fa\u9519\u7387\u8fd8\u662f\u633a\u4f4e\u7684
\n\u9898\u76ee\u94fe\u63a5\uff1a
\n\u6c34\u9898<\/a>
\n\u82e5\u5e72\u6d4b\u8bd5\u6570\u636e\uff0c\u6bcf\u884c\u4e00\u4e2a\u4e00\u767e\u4f4d\u4ee5\u5185\u7684\u6b63\u6574\u6570\uff0c\u5224\u65ad\u662f\u5426\u4e3a\u7d20\u6570\u3002("Yes" "No")
\n\u6570\u8bba\u4e00\u00b7Miller-Rabin\u8d28\u6570\u6d4b\u8bd5<\/a>
\n\u7b2c\u4e00\u884c\u7ed9\u51fa\u6d4b\u8bd5\u6570\u636e\u4e2a\u6570\uff0c\u63a5\u4e0b\u6765\u6bcf\u884c\u4e00\u4e2along long\u8303\u56f4\u5185\u7684\u6570\uff0c\u6d4b\u8bd5\u662f\u5426\u4e3a\u7d20\u6570\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"