{"id":305,"date":"2026-03-16T20:24:28","date_gmt":"2026-03-16T12:24:28","guid":{"rendered":"https:\/\/jiangqvweihuan.cn\/?p=305"},"modified":"2026-03-16T20:24:29","modified_gmt":"2026-03-16T12:24:29","slug":"%e7%b4%a0%e6%95%b0%e5%88%a4%e5%ae%9a%e5%8f%8a%e7%b4%a0%e6%95%b0%e7%ad%9b","status":"publish","type":"post","link":"https:\/\/jiangqvweihuan.cn\/index.php\/2026\/03\/16\/%e7%b4%a0%e6%95%b0%e5%88%a4%e5%ae%9a%e5%8f%8a%e7%b4%a0%e6%95%b0%e7%ad%9b\/","title":{"rendered":"\u7b97\u6cd5\u5fc3\u4f20\u00b7\u7d20\u6570\u5224\u5b9a\u53ca\u7d20\u6570\u7b5b"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">\u4e00\u3001\u7d20\u6570\uff08\u8d28\u6570\uff09<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u5f15\u5165<\/h3>\n\n\n\n<p>\u7d20\u6570\u53c8\u79f0\u8d28\u6570\uff0c\u662f\u5927\u4e8e1\u7684\u81ea\u7136\u6570\u4e2d\u53ea\u80fd\u88ab1\u548c\u5b83\u672c\u8eab\u6574\u9664\u7684\u6570\u3002\u76f8\u5bf9\u7684\uff0c\u5982\u679c\u4e00\u4e2a\u5927\u4e8e1\u7684\u81ea\u7136\u6570\u80fd\u88ab\u9664\u4e861\u548c\u5b83\u672c\u8eab\u4ee5\u5916\u7684\u6574\u6570\u6574\u9664\u7684\u6570\u53eb\u505a\u5408\u6570\u3002\u7279\u522b\u7684\uff0c1\u65e2\u4e0d\u662f\u8d28\u6570\u4e5f\u4e0d\u662f\u5408\u6570\u3002<\/p>\n\n\n\n<p>\u7d20\u6570\u6709\u7740\u5305\u62ec\u4ee5\u4e0b\u51e0\u6761\u6709\u8da3\u7684\u4e8b\u5b9e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7d20\u6570\u7684\u6570\u91cf\u6709\u65e0\u9650\u591a\u3002<\/li>\n\n\n\n<li>\u7d20\u6570\u7684\u5206\u5e03\u968f\u7740\u6574\u6570\u7684\u589e\u5927\u5206\u5e03\u8d8a\u6765\u8d8a\u7a00\u758f\u3002<\/li>\n\n\n\n<li>\u5bf9\u4e8e\u4efb\u610f\u6b63\u6574\u6570n\uff0c\u81f3\u5c11\u5b58\u5728n\u4e2a\u8fde\u7eed\u7684\u6b63\u5408\u6570\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u540c\u65f6\uff0c\u6709\u5927\u91cf\u5173\u4e8e\u7d20\u6570\u7684\u731c\u60f3\uff0c\u4ee5\u4e0b\u662f\u51e0\u4e2a\u8457\u540d\u7684\u731c\u60f3\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6ce2\u7279\u5170\u731c\u60f3\uff1a\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a\u7ed9\u5b9a\u7684\u6b63\u6574\u6570n>1\uff0c\u5b58\u5728\u4e00\u4e2a\u7d20\u6570p\uff0c\u4f7f\u5f97n&lt;p&lt;2n\u3002\u5df2\u7ecf\u8bc1\u660e\u3002<\/li>\n\n\n\n<li>\u5b6a\u751f\u7d20\u6570\u731c\u60f3\uff1a\u5b58\u5728\u65e0\u7a77\u591a\u4e2a\u5f62\u5982p\u548cp+2\u7684\u7d20\u6570\u5bf9\u3002\u5c1a\u672a\u8bc1\u660e\u3002<\/li>\n\n\n\n<li>\u7d20\u6570\u7b49\u5dee\u6570\u5217\u731c\u60f3\uff1a\u5bf9\u4e8e\u4efb\u610f\u6b63\u6574\u6570n>2\uff0c\u6709\u4e00\u4e2a\u7531\u7d20\u6570\u7ec4\u6210\u7684\u957f\u5ea6\u4e3an\u7684\u7b49\u5dee\u6570\u5217\u3002\u5df2\u8bc1\u660e\u5b58\u5728\u4efb\u610f\u957f\u7684\u7d20\u6570\u7b49\u5dee\u6570\u5217\u3002<\/li>\n\n\n\n<li>\u54e5\u5fb7\u5df4\u8d6b\u731c\u60f3\uff1a\u6bcf\u4e2a\u5927\u4e8e2\u7684\u6b63\u5076\u6570\u53ef\u4ee5\u5199\u6210\u4e24\u4e2a\u7d20\u6570\u7684\u548c\u3002\u6700\u6709\u540d\u7684\u7d20\u6570\u731c\u60f3\uff0c\u5c1a\u672a\u8bc1\u660e\u3002<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e8c\u3001\u5355\u7d20\u6570\u5224\u5b9a<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u7d20\u6570\u7684\u5224\u5b9a<\/h3>\n\n\n\n<p>\u7d20\u6570\u7684\u5224\u5b9a\u65b9\u6cd5\u4e3b\u8981\u6709\u4e09\u79cd\uff0c\u7b2c\u4e00\u79cd\u662f\u8bd5\u9664\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u7d20\u6570\u7684\u5b9a\u4e49\u4ece2\u9010\u4e2a\u7f57\u5217\uff0c\u8bd5\u7740\u53bb\u6574\u9664n\uff0c\u5982\u679cn\u53ef\u4ee5\u88ab\u5176\u4e2d\u4efb\u4f55\u4e00\u4e2a\u6570\u6574\u9664\u5c31\u8bf4\u660e\u5176\u4e0d\u662f\u7d20\u6570\u3002\u5f88\u5bb9\u6613\u53d1\u73b0\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u8303\u56f4\u4ece[2\uff0cn-1]\u7f29\u5c0f\u5230[2\uff0csqrt(n)]\u3002\u4e0b\u9762\u7ed9\u51fa\u89e3\u51b3\u95ee\u9898\u7684\u51fd\u6570\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\/\/ ll \u4ee3\u66ff long long \u8be6\u89c1\u7ade\u8d5b\u6a21\u677f\nbool prime(ll a){\n    if(a &lt; 2)\n        return false;\n    for (int i = 2; i &lt;= sqrt(a);++i)\n        if(a % i == 0)\n            return false;\n    return true;\n}<\/code><\/pre>\n\n\n\n<p>\u4ee3\u7801\u5f88\u597d\u7406\u89e3\uff0c\u8fd9\u91cc\u4e0d\u8fc7\u591a\u89e3\u91ca\u3002\u4e0d\u96be\u53d1\u73b0\uff0c\u5f53a\u5c0f\u4e8e1e12\u65f6\uff0c\u5f00\u65b9\u7684\u590d\u6742\u5ea6\u662f\u53ef\u4ee5\u8fd0\u884c\u4e0d\u8d85\u65f6\u7684\uff0c\u4f46\u662f\u968f\u7740a\u7684\u8fdb\u4e00\u6b65\u53d8\u5927\uff0c\u7d20\u6570\u7684\u5224\u5b9a\u7528\u8bd5\u9664\u6cd5\u5c31\u6709\u70b9\u4e0d\u591f\u7528\u4e86\uff0c\u8fd9\u65f6\u5019\u5c31\u8981\u5f15\u51fa\u4e0b\u9762\u7684\u4e24\u79cd\u65b9\u6cd5\u4e86\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.Fermat \u7d20\u6027\u6d4b\u8bd5<\/h3>\n\n\n\n<p>Fermat \u7d20\u6027\u6d4b\u8bd5\u4e2d\u6587\u540d\u53eb\u8d39\u9a6c\u7d20\u6027\u6d4b\u8bd5\uff0c\u4ed6\u57fa\u4e8e\u8d39\u9a6c\u5c0f\u5b9a\u7406\u3002<\/p>\n\n\n\n<p>\u8d39\u9a6c\u5c0f\u5b9a\u7406\uff1a\u8bben\u662f\u7d20\u6570\uff0ca\u662f\u6b63\u6574\u6570\u4e14\u4e0en\u4e92\u7d20\uff0c\u90a3\u4e48\u6709a<sup>n-1<\/sup>\u22611(mod n)\u3002\u8d39\u9a6c\u5c0f\u5b9a\u7406\u7684\u9006\u547d\u9898\u4e5f\u51e0\u4e4e\u6210\u7acb\uff0cFermat \u7d20\u6027\u6d4b\u8bd5\u5c31\u662f\u57fa\u4e8e\u8d39\u9a6c\u5c0f\u5b9a\u7406\u7684\u9006\u547d\u9898\uff0c\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u65b9\u6cd5\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u6d4b\u8bd5n\u662f\u5426\u4e3a\u7d20\u6570\uff0c\u57281~n\u4e4b\u95f4\u4efb\u9009\u4e00\u4e2a\u968f\u673a\u7684\u57fa\u503ca\uff0ca\u5e76\u4e0d\u4e00\u5b9a\u8981\u548cn\u4e92\u7d20\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5982\u679ca<sup>n-1<\/sup>\u22611(mod n)\u4e0d\u6210\u7acb\uff0c\u90a3\u4e48n\u80af\u5b9a\u4e0d\u662f\u7d20\u6570\u3002<\/li>\n\n\n\n<li>\u5982\u679ca<sup>n-1<\/sup>\u22611(mod n)\u6210\u7acb\uff0c\u90a3\u4e48n\u5927\u6982\u7387\u662f\u7d20\u6570\uff0c\u5c1d\u8bd5\u7684a\u8d8a\u591a\uff0cn\u662f\u7d20\u6570\u7684\u6982\u7387\u8d8a\u5927\uff0c\u6211\u4eec\u79f0n\u662f\u4e00\u4e2a\u57fa\u4e8ea\u7684\u4f2a\u7d20\u6570\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u4f46\u662f\uff0c\u53ef\u60dc\u7684\u662f\u4ece\u7b2c\u4e8c\u79cd\u60c5\u51b5\u53ef\u4ee5\u770b\u51faFermat \u7d20\u6027\u6d4b\u8bd5\u5e76\u4e0d\u662f\u5b8c\u5168\u6b63\u786e\u7684\u3002\u5bf9\u4e8e\u67d0\u4e2aa\u7684\u503c\uff0c\u603b\u6709\u4e00\u4e9b\u5408\u6570\u88ab\u8bef\u5224\u901a\u8fc7\u6d4b\u8bd5\u3002\u4e0d\u540c\u7684a\u503c\uff0c\u88ab\u8bef\u5224\u7684\u5408\u6570\u90fd\u4e0d\u592a\u4e00\u6837\uff0c\u4f46\u662f\u7279\u522b\u7684\uff0c\u603b\u6709\u4e00\u90e8\u5206\u5408\u6570\u65e0\u8bbaa\u600e\u4e48\u9009\uff0c\u90fd\u80fd\u901a\u8fc7\u6d4b\u8bd5\uff0c\u6211\u4eec\u79f0\u5176\u4e3aCarminchsel\u6570\uff0c\u4e2d\u6587\u540d\u53eb\u505a\u5361\u8fc8\u514b\u5c14\u6570\u3002\u524d\u4e09\u4e2aCarminchsel\u6570\u662f561\uff0c1105\u548c1729\u3002Carminchsel\u6570\u5f88\u5c11\uff0c\u524d1\u4ebf\u7684\u6570\u91cc\u9762\u4e5f\u5c31\u53ea\u6709255\u4e2a\uff0c\u5e76\u4e14\u5f53n\u8d8b\u5411\u4e8e\u65e0\u7a77\u65f6\uff0cCarminchsel\u6570\u7684\u5206\u5e03\u6781\u5176\u7a00\u758f\uff0cFermat \u7d20\u6027\u6d4b\u8bd5\u51e0\u4e4e\u4e0d\u4f1a\u51fa\u9519\uff0c\u6240\u4ee5\u5b83\u4e00\u79cd\u76f8\u5f53\u4e0d\u9519\u7684\u65b9\u6cd5\uff0c\u4e0b\u9762\u7ed9\u51fa\u5b83\u7684\u5b9e\u73b0\u51fd\u6570\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ll mul(ll a, ll b, ll mod)\n{\n    \/\/ return (ll)((__int128)a * b % mod);  \/\/ \u73af\u5883\u5141\u8bb8\u9009\u62e9\u8fd9\u4e00\u6761\u8bed\u53e5\n    \/\/ \u5148\u5c06 a \u548c b \u5bf9 mod \u53d6\u6a21\uff0c\u7f29\u5c0f\u8303\u56f4\uff08\u4f46\u82e5 mod \u5f88\u5927\uff0c\u4ecd\u53ef\u80fd\u6ea2\u51fa\uff09\n    a %= mod;\n    b %= mod;\n\n    ll res = 0; \/\/ \u7d2f\u52a0\u7ed3\u679c\n    while (b > 0)\n    {\n        if (b &amp; 1)\n        {                          \/\/ \u5982\u679c\u5f53\u524d\u4e8c\u8fdb\u5236\u4f4d\u4e3a1\uff0c\u5219\u7d2f\u52a0\u5f53\u524d\u7684 a\n            res = (res + a) % mod; \/\/ \u6ce8\u610f\uff1ares + a \u53ef\u80fd\u6ea2\u51fa\uff08\u5f53 mod \u63a5\u8fd1 LLONG_MAX\uff09\n        }\n        a = (a + a) % mod; \/\/ a \u7ffb\u500d\uff0c\u5bf9\u5e94\u4e8c\u8fdb\u5236\u4f4d\u7684\u6743\u91cd\n        b >>= 1;           \/\/ \u5904\u7406\u4e0b\u4e00\u4f4d\n    }\n    return res;\n}\n\nll qpow(ll a, ll b, ll mod)\n{\n    ll res = 1;\n    a %= mod;\n    while (b)\n    {\n        if (b &amp; 1)\n            res = mul(res, a, mod); \/\/ \u5f53\u524d\u4f4d\u4e3a1\u65f6\u7d2f\u4e58\n        a = mul(a, a, mod);         \/\/ \u5e95\u6570\u5e73\u65b9\n        b >>= 1;                    \/\/ \u6307\u6570\u53f3\u79fb\n    }\n    return res;\n}\n\nbool fermat(ll n, int k = 50)\n{\n    if (n &lt; 2)\n        return false;\n    for (int i = 0; i &lt; k; ++i)\n    {\n        ll a = rand() % (n - 1) + 1; \/\/ \u968f\u673a\u9009\u53d6\u968f\u673a\u6570\uff0c\u8303\u56f4\u4e3a1\u5230n-1\n        if (qpow(a, n - 1, n) != 1)\n            return false;\n    }\n    return true;\n}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">3.Miller-Rabin\u7d20\u6027\u6d4b\u8bd5<\/h3>\n\n\n\n<p>Miller-Rabin\u7d20\u6027\u6d4b\u8bd5\u4e2d\u6587\u540d\u53eb\u7c73\u52d2-\u62c9\u5bbe\u7d20\u6027\u6d4b\u8bd5\uff0c\u53ef\u4ee5\u770b\u4f5c\u662fFermat\u7d20\u6027\u6d4b\u8bd5\u7684\u6539\u8fdb\u7248\uff0c\u53ef\u4ee5\u7528\u4e8e\u5224\u65ad\u6392\u9664Carminchsel\u6570\u3002Miller-Rabin\u7d20\u6027\u6d4b\u8bd5\u7528\u5230\u4e86\u4e8c\u6b21\u63a2\u6d4b\u5b9a\u7406\uff0c\u5982\u679cp\u662f\u4e00\u4e2a\u5947\u7d20\u6570\uff0c\u4e14e>=1,\u5219\u65b9\u7a0bx<sup>2<\/sup>\u22611(mod p<sup>e<\/sup>)\u4ec5\u6709\u4e24\u4e2a\u89e3\uff1ax=1\u548cx=-1\u3002\u5f53e=1\u65f6\uff0c\u65b9\u7a0bx<sup>2<\/sup>\u22611(mod p)\u4ec5\u6709\u4e24\u4e2a\u89e3\uff0cx=1\u548cx=p-1\u3002\u6211\u4eec\u628ax=1\u548cx=p-1\u79f0\u4f5c\u201cx\u5bf9\u6a21p\u6765\u8bf41\u7684\u5e73\u51e1\u5e73\u65b9\u6839\u201d\u3002\u4f8b\u5982x=6\uff0cn=35\uff0c6\u662f\u5bf9\u6a2135\u6765\u8bf4\u76841\u7684\u5e73\u51e1\u5e73\u65b9\u6839\u3002<\/p>\n\n\n\n<p>\u7531\u4e8c\u6b21\u63a2\u6d4b\u5b9a\u7406\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u5982\u679c\u5bf9\u6a21n\u5b58\u57281\u7684\u975e\u5e73\u51e1\u5e73\u65b9\u6839\uff0c\u90a3\u4e48n\u662f\u5408\u6570\u3002\u8fd9\u4e2a\u63a8\u8bba\u5c31\u662f\u4e8c\u6b21\u63a2\u6d4b\u5b9a\u7406\u7684\u9006\u5426\u547d\u9898\uff0c\u5373\u5982\u679c\u5bf9n\u5b58\u57281\u7684\u975e\u5e73\u51e1\u5e73\u65b9\u6839\uff0c\u5219n\u4e0d\u53ef\u80fd\u662f\u5947\u7d20\u6570\u6216\u8005\u5947\u7d20\u6570\u7684\u5e42\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u5bf9\u4e4b\u524d\u7684Fermat\u7d20\u6570\u6d4b\u8bd5\u7684\u51fd\u6570\u505a\u51fa\u8c03\u6574\uff0c\u5982\u679ca<sup>n-1<\/sup>\u22611(mod n)\u4e0d\u6210\u7acb\uff0c\u90a3\u4e48n\u80af\u5b9a\u4e0d\u662f\u7d20\u6570\uff0c\u7531\u4e8en-1\u5b9e\u5728\u592a\u5927\uff0c\u663e\u7136\u76f4\u63a5\u8ba1\u7b97a<sup>n-1<\/sup>\u4e0d\u53ef\u884c\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5230\u6570\u5b66\u6280\u5de7\uff1a\u628an-1\u8868\u793a\u4e3a\u5e42\u7684\u5f62\u5f0f\uff0c\u7136\u540e\u501f\u52a9\u5feb\u901f\u5e42\u6765\u8fd0\u7b97\u3002<\/p>\n\n\n\n<p>\u4ee4n-1 = 2<sup>t<\/sup>u\uff0c\u5176\u4e2du\u662f\u5947\u6570\uff0ct\u662f\u6b63\u6574\u6570\u3002\u5982\u4f55\u6c42t\u548cu\u5462\uff1f\u8fd9\u91cc\u76f4\u63a5\u7ed9\u51fa\u65b9\u6848\uff1an-1\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u662f\u5947\u6570u\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u540e\u9762\u52a0t\u4e2a0\u3002\u5982255-1 = 2<sup>5<\/sup>*7\uff0cn-1=255-1=244\u7684\u4e8c\u8fdb\u5236\u662f11100000\uff0c\u5b83\u662fu=7\u4e8c\u8fdb\u5236111\uff0c\u540e\u9762\u52a0t\u4e2a0\u3002<\/p>\n\n\n\n<p>\u6545\u800c\u516c\u5f0f\u8f6c\u5316\u4e3a\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center\">a<sup>n-1<\/sup>\u2261(a<sup>u<\/sup>)<math data-latex=\"^{2^t}\"><semantics><msup><mrow><\/mrow><msup><mn>2<\/mn><mi>t<\/mi><\/msup><\/msup><annotation encoding=\"application\/x-tex\">^{2^t}<\/annotation><\/semantics><\/math>(mod n)<\/p>\n\n\n\n<p>\u5f53Miller-Rabin\u7d20\u6027\u6d4b\u8bd5\u6b21\u6570\u8fbe\u523050\u6b21\u65f6\uff0c\u51fa\u9519\u6982\u7387\u5df2\u7ecf\u5c0f\u5230\u53ef\u4ee5\u5ffd\u89c6\u4e86\uff0c\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u63a8\u5bfcCarminchsel\u6570561\u662f\u5426\u80fd\u88abMiller-Rabin\u7d20\u6027\u6d4b\u8bd5\u6392\u9664\u6389\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u60f3\u8981\u8be6\u7ec6\u4e86\u89e3\u539f\u7406\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u53bb\u641c\u7d22\u5b66\u4e60\u8bc1\u660e\u8fc7\u7a0b\uff0c\u8fd9\u91cc\u76f4\u63a5\u7ed9\u51fa\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ll mul(ll a, ll b, ll mod)\n{\n    \/\/ return (ll)((__int128)a * b % mod);  \/\/ \u73af\u5883\u5141\u8bb8\u9009\u62e9\u8fd9\u4e00\u6761\u8bed\u53e5\n    \/\/ \u5148\u5c06 a \u548c b \u5bf9 mod \u53d6\u6a21\uff0c\u7f29\u5c0f\u8303\u56f4\uff08\u4f46\u82e5 mod \u5f88\u5927\uff0c\u4ecd\u53ef\u80fd\u6ea2\u51fa\uff09\n    a %= mod;\n    b %= mod;\n\n    ll res = 0; \/\/ \u7d2f\u52a0\u7ed3\u679c\n    while (b > 0)\n    {\n        if (b &amp; 1)\n        {                          \/\/ \u5982\u679c\u5f53\u524d\u4e8c\u8fdb\u5236\u4f4d\u4e3a1\uff0c\u5219\u7d2f\u52a0\u5f53\u524d\u7684 a\n            res = (res + a) % mod; \/\/ \u6ce8\u610f\uff1ares + a \u53ef\u80fd\u6ea2\u51fa\uff08\u5f53 mod \u63a5\u8fd1 LLONG_MAX\uff09\n        }\n        a = (a + a) % mod; \/\/ a \u7ffb\u500d\uff0c\u5bf9\u5e94\u4e8c\u8fdb\u5236\u4f4d\u7684\u6743\u91cd\n        b >>= 1;           \/\/ \u5904\u7406\u4e0b\u4e00\u4f4d\n    }\n    return res;\n}\n\nll qpow(ll a, ll b, ll mod)\n{\n    ll res = 1;\n    a %= mod;\n    while (b)\n    {\n        if (b &amp; 1)\n            res = mul(res, a, mod); \/\/ \u5f53\u524d\u4f4d\u4e3a1\u65f6\u7d2f\u4e58\n        a = mul(a, a, mod);         \/\/ \u5e95\u6570\u5e73\u65b9\n        b >>= 1;                    \/\/ \u6307\u6570\u53f3\u79fb\n    }\n    return res;\n}\n\nbool witness(ll a, ll n)\n{\n    \/\/ \u5c06 n-1 \u5206\u89e3\u4e3a u * 2^t\uff0c\u5176\u4e2d u \u4e3a\u5947\u6570\n    ll u = n - 1;\n    int t = 0;\n    while ((u &amp; 1) == 0) \/\/ \u5f53 u \u4e3a\u5076\u6570\u65f6\n    {\n        u = u >> 1; \/\/ \u4e0d\u65ad\u9664\u4ee5 2\n        t++;        \/\/ \u8bb0\u5f55\u9664\u4ee5 2 \u7684\u6b21\u6570\n    }\n\n    \/\/ \u8ba1\u7b97 a^u mod n\uff0c\u4f5c\u4e3a\u521d\u59cb\u503c\n    ll x1 = qpow(a, u, n); \/\/ x1 = a^u % n\n    ll x2;                 \/\/ \u7528\u4e8e\u5b58\u50a8\u5e73\u65b9\u540e\u7684\u503c\n\n    \/\/ \u8fdb\u884c t \u6b21\u5e73\u65b9\uff0c\u4f9d\u6b21\u68c0\u67e5\u662f\u5426\u51fa\u73b0\u975e\u5e73\u51e1\u5e73\u65b9\u6839\n    for (int i = 1; i &lt;= t; ++i)\n    {\n        x2 = mul(x1, x1, n); \/\/ x2 = x1^2 % n\n        if (x2 == 1 &amp;&amp; x1 != 1 &amp;&amp; x1 != n - 1)\n            return true; \/\/ \u53d1\u73b0\u975e\u5e73\u51e1\u5e73\u65b9\u6839 \u2192 n \u662f\u5408\u6570\n        x1 = x2;         \/\/ \u66f4\u65b0 x1 \u7ee7\u7eed\u4e0b\u4e00\u8f6e\u5e73\u65b9\n    }\n\n    \/\/ \u7ecf\u8fc7 t \u6b21\u5e73\u65b9\u540e\uff0c\u6700\u7ec8\u503c\u5e94\u4e3a 1\uff08\u82e5 n \u662f\u7d20\u6570\u4e14 a \u4e0e n \u4e92\u8d28\uff09\n    if (x1 != 1)\n        return true; \/\/ \u6700\u7ec8\u503c\u4e0d\u4e3a 1 \u2192 n \u662f\u5408\u6570\n    return false;    \/\/ \u672c\u6b21\u6d4b\u8bd5\u672a\u53d1\u73b0\u5408\u6570\u8bc1\u636e\uff0cn \u53ef\u80fd\u662f\u7d20\u6570\n}\n\nbool miller_rabin(ll n)\n{\n    int s = 50; \/\/ \u6d4b\u8bd5\u6b21\u6570\uff0c\u53ef\u6839\u636e\u9700\u8981\u8c03\u6574\n    if (n &lt; 2)\n        return false; \/\/ \u5c0f\u4e8e 2 \u7684\u6570\u4e0d\u662f\u7d20\u6570\n    if (n == 2)\n        return true; \/\/ 2 \u662f\u7d20\u6570\n    if (n % 2 == 0)\n        return false; \/\/ \u5076\u6570\uff08\u9664 2 \u5916\uff09\u4e0d\u662f\u7d20\u6570\n\n    \/\/ \u8fdb\u884c s \u6b21\u72ec\u7acb\u6d4b\u8bd5\n    for (int i = 0; i &lt; s; ++i)\n    {\n        \/\/ \u968f\u673a\u9009\u53d6\u5e95\u6570 a\uff0c\u8303\u56f4 &#91;1, n-1]\n        ll a = rand() % (n - 1) + 1;\n        if (witness(a, n)) \/\/ \u82e5 witness \u8fd4\u56de true\uff0c\u8bf4\u660e n \u662f\u5408\u6570\n            return false;\n    }\n    return true; \/\/ \u6240\u6709\u6d4b\u8bd5\u901a\u8fc7\uff0cn \u5f88\u53ef\u80fd\u662f\u7d20\u6570\n}<\/code><\/pre>\n\n\n\n<p>\u7d20\u6027\u68c0\u6d4b\u80fd\u5904\u7406\u7684\u6700\u5927\u53d8\u91cflong long\u7ea6\u4e3a1e19\uff0c\u518d\u5927\u5c31\u8981\u81ea\u5df1\u5904\u7406\u9ad8\u7cbe\u5ea6\u5927\u6570\u4e86\uff0c\u4e0b\u9762\u6211\u4eec\u5c06\u6df1\u5165\u63a2\u7a76\u5982\u4f55\u5927\u6279\u91cf\u5224\u65ad\u8fde\u7eed\u7684\u4e00\u6bb5\u6570\u5b57\u4e2d\u7684\u7d20\u6570\u7684\u4e2a\u6570\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e09\u3001\u7d20\u6570\u7b5b<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u5f15\u5165<\/h3>\n\n\n\n<p>\u60f3\u60f3\u4ee5\u4e0b\u4ee5\u4e0b\u573a\u666f\uff1a\u7ed9\u5b9a\u4f60\u4e00\u4e2an\uff0c\u6c422~n\u4e4b\u95f4\u7684\u6240\u6709\u7d20\u6570\u3002<\/p>\n\n\n\n<p>\u4e0d\u96be\u53d1\u73b0\uff0c\u5982\u679c\u9010\u4e2a\u5224\u65ad\uff0c\u5f53n\u6781\u5927\u65f6\uff0c\u663e\u7136\u4f1a\u8d85\u65f6\uff0c\u8fd9\u65f6\u5019\u5c31\u8981\u7528\u7d20\u6570\u7b5b\u7b5b\u9009\u6240\u6709\u6574\u6570\uff0c\u628a\u975e\u8d1f\u6570\u7b5b\u53bb\u3002\u5e38\u89c1\u7684\u7d20\u6570\u7b5b\u6709\u4e24\u79cd\uff0c\u57c3\u6c0f\u7b5b\u548c\u6b27\u62c9\u7b5b\uff0c\u6b27\u62c9\u7b5b\u7684\u4e8b\u4ef6\u590d\u6742\u5ea6\u65f6O(n)\uff0c\u8fbe\u5230\u6700\u4f18\uff0c\u4e0d\u53ef\u80fd\u66f4\u5feb\u4e86\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.\u57c3\u6c0f\u7b5b<\/h3>\n\n\n\n<p>\u57c3\u6c0f\u7b5b\uff08Eratosthenes\u00a0Sieve\uff09\u4e2d\u6587\u5168\u79f0\u57c3\u62c9\u6258\u65af\u7279\u5c3c\u7b5b\u6cd5\uff0c\u5b83\u7528\u7684\u662f\u7d20\u6570\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u5bf9\u521d\u59cb2~n\u7684\u961f\u5217\uff08\u9ed8\u8ba4\u4ece\u5c0f\u5230\u5927\uff09\u8fdb\u884c\u7b5b\u9009\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u8f93\u51fa\u6700\u5c0f\u7684\u7d20\u65702\uff0c\u7136\u540e\u7b5b\u6389\u6240\u67092\u7684\u500d\u6570\u3002<\/li>\n\n\n\n<li>\u8f93\u51fa\u6700\u5c0f\u7684\u7d20\u65703\uff0c\u7136\u540e\u7b5b\u6389\u6240\u67093\u7684\u500d\u6570\u3002<\/li>\n\n\n\n<li>\u8f93\u51fa\u6700\u5c0f\u7684\u7d20\u65705\uff0c\u7136\u540e\u7b5b\u6389\u6240\u67095\u7684\u500d\u6570\u3002<\/li>\n\n\n\n<li>&#8230;&#8230;<\/li>\n<\/ol>\n\n\n\n<p>\u91cd\u590d\u4ee5\u4e0a\u6b65\u9aa4\uff0c\u76f4\u5230\u6700\u540e\u961f\u5217\u4e3a\u7a7a\u3002<\/p>\n\n\n\n<p>\u4e0b\u9762\u7ed9\u51fa\u4ee3\u7801\uff0c\u65b9\u4fbf\u8bfb\u8005\u7406\u89e3\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>int E_sieve(ll n)\n{\n    vector&lt;bool> vis(n + 1, false);\n    vector&lt;int> prime;\n    for (int i = 2; i &lt;= sqrt(n); ++i) \/\/ \u539f\u7406\u548c\u8bd5\u9664\u6cd5\u76f8\u4f3c\n    {\n        if (!vis&#91;i])\n            for (int j = i * i; j &lt;= n; j += i)\n                vis&#91;j] = true;\n    }\n    for (int i = 2; i &lt;= n; ++i)\n        if (!vis&#91;i])\n            prime&#91;k++] = i;\n    return prime.size();\n}<\/code><\/pre>\n\n\n\n<p>\u8981\u662f\u8bfb\u8005\u60f3\u4fdd\u5b58\u6bcf\u4e00\u4e2a\u7d20\u6570\u53ef\u4ee5\u5c06\u6570\u7ec4\u5b9a\u4e49\u5728\u5168\u5c40\u91cc\u3002\u76f8\u8f83\u4e8e\u9a6c\u4e0a\u8981\u4ecb\u7ecd\u7684\u6b27\u62c9\u7b5b\uff0c\u57c3\u6c0f\u7b5b\u8fd8\u662f\u7565\u6709\u4e0d\u8db3\uff0c\u6bd4\u598212\u88ab2\u548c3\u7b5b\u4e86\u4e24\u6b21\uff0c\u5177\u4f53\u6027\u80fd\u6bd4\u8f83\u8bfb\u8005\u53ef\u81ea\u884c\u67e5\u9605\u76f8\u5173\u8d44\u6599\uff0c\u5728\u6b64\u4e0d\u505a\u8be6\u7ec6\u5206\u6790\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.\u6b27\u62c9\u7b5b<\/h3>\n\n\n\n<p>\u6b27\u62c9\u7b5b\uff08Euler Sieve\uff09\uff0c\u662f\u57c3\u6c0f\u7b5b\u7684\u6539\u826f\u7248\uff0c\u8fbe\u5230\u4e86\u7ebf\u6027\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u6b64\u53c8\u79f0\u7ebf\u6027\u7b5b\u3002<\/p>\n\n\n\n<p>\u6b27\u62c9\u7b5b\u7684\u539f\u7406\uff1a\u4e00\u4e2a\u5408\u6570\u80af\u5b9a\u6709\u4e00\u4e2a\u6700\u5c0f\u8d28\u56e0\u6570\uff0c\u8ba9\u6bcf\u4e2a\u5408\u6570\u53ea\u88ab\u5b83\u7684\u6700\u5c0f\u8d28\u56e0\u6570\u7b5b\u9009\u4e00\u6b21\uff0c\u4ee5\u8fbe\u5230\u4e0d\u91cd\u590d\u7b5b\u9009\u7684\u76ee\u7684\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u539f\u7406\u5982\u4e0b\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u9010\u4e00\u68c0\u67e52~n\u7684\u6240\u6709\u6570\uff0c\u7b2c\u4e00\u4e2a\u68c0\u67e5\u7684\u662f2\uff0c\u5b83\u662f\u7b2c\u4e00\u4e2a\u7d20\u6570\u3002<\/li>\n\n\n\n<li>\u5f53\u68c0\u67e5\u5230\u7b2ci\u4e2a\u6570\u7684\u65f6\uff0c\u5229\u7528\u5df2\u7ecf\u6c42\u5f97\u7684\u7d20\u6570\u7b5b\u53bb\u5bf9\u5e94\u7684\u5408\u6570x\uff0c\u800c\u4e14\u662f\u7528x\u7684\u6700\u5c0f\u8d28\u56e0\u6570\u53bb\u7b5b\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u4e0b\u9762\u7ed9\u51fa\u6b27\u62c9\u7b5b\u7684\u4ee3\u7801\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>vector&lt;ll> euler_sieve(ll n)\n{\n    vector&lt;bool> vis(n + 1, true); \/\/ \u6807\u8bb0\u662f\u5426\u4e3a\u7d20\u6570\uff0c\u521d\u59cb\u5047\u8bbe\u5168\u662f\u7d20\u6570\n    vector&lt;ll> prime;              \/\/ \u5b58\u50a8\u5df2\u53d1\u73b0\u7684\u7d20\u6570\n\n    for (ll i = 2; i &lt;= n; ++i)\n    {\n        if (vis&#91;i])\n        { \/\/ i \u662f\u7d20\u6570\uff0c\u52a0\u5165\u5217\u8868\n            prime.push_back(i);\n        }\n        ll size = prime.size();\n        \/\/ \u7528 i \u4e58\u4ee5\u6bcf\u4e2a\u5df2\u53d1\u73b0\u7684\u7d20\u6570\u6765\u7b5b\u9664\u5408\u6570\n        for (ll j = 0; j &lt; size; ++j)\n        {\n            if (i > n \/ prime&#91;j])\n                break;\n            vis&#91;i * prime&#91;j]] = false; \/\/ \u6807\u8bb0\u4e3a\u5408\u6570\n            \/\/ \u5982\u679c primes&#91;j] \u662f i \u7684\u6700\u5c0f\u8d28\u56e0\u5b50\uff0c\u5219\u505c\u6b62\u5faa\u73af\n            if (i % prime&#91;j] == 0)\n                break;\n        }\n    }\n    return prime;\n}<\/code><\/pre>\n\n\n\n<p>\u867d\u7136\u6bd4\u8d5b\u53ef\u80fd\u7528\u4e0d\u5230long long\u8303\u56f4\uff0c\u4f46\u662f\u4e8e\u6b64\u77e5\u8bc6\u4e3a\u4e86\u656c\u544a\u8bfb\u8005\uff0c\u8b66\u949f\u957f\u9e23\uff01\uff01\uff01<\/p>\n\n\n\n<p>\u8fd9\u91cc\u6709\u4e00\u4e2a\u5c0f\u6280\u5de7\uff0c\u5728\u4e00\u822c\u7b97\u6cd5\u7ade\u8d5b\u73af\u5883\u4e2d\u7684\u7a7a\u95f4\u9650\u5236\u4e0b\u6211\u4eec\u5982\u679c\u8981\u4f7f\u7528\u9759\u6001\u6570\u7ec4\uff0c\u6700\u5927\u53ef\u4ee5\u5f00\u5230int prime[5800000]\u6765\u50a8\u5b58N=1e8\u8303\u56f4\u5185\u7684\u6240\u6709\u7d20\u6570\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u7ed9\u5230\u4e00\u4e2a\u62d3\u5c55\uff0c\u6c421~n\u4e2d\u6bcf\u4e00\u4e2a\u6570\u7684\u6700\u5c0f\u8d28\u56e0\u6570\uff0c\u8fd9\u91cc\u53ea\u9700\u8981\u7b80\u5355\u7684\u5bf9\u6807\u8bb0\u6570\u7ec4\u4fee\u6539\u53d8\u91cf\u7c7b\u578b\u4e3aint\u7136\u540e\u6bcf\u6b21\u50a8\u5b58\u7684\u5c31\u662f\u6b27\u62c9\u7b5b\u7b2c\u4e00\u6b21\u7b5b\u5230\u8fd9\u4e2a\u5408\u6570\u7684\u7d20\u6570\u503c\uff0c\u5bf9\u4e8e\u7d20\u6570\uff0c\u6700\u5c0f\u8d28\u56e0\u6570\u5c31\u662f\u5b83\u672c\u8eab\uff0c\u8fd9\u4e2a\u95ee\u9898\u5c31\u89e3\u51b3\u4e86\uff0c\u8fd9\u91cc\u4e0d\u7ed9\u51fa\u4ee3\u7801\uff0c\u671b\u8bfb\u8005\u80fd\u591f\u81ea\u884c\u601d\u8003\uff0c\u81ea\u5df1\u5b8c\u6210\u8be5\u4ee3\u7801\u7136\u540e\u53bb\u6d4b\u8bd5\uff0c\u5982\u679c\u8fd8\u662f\u6709\u70b9\u4e0d\u592a\u7406\u89e3\u6211\u8bf4\u7684\u8fd9\u4e00\u6bb5\u8bdd\uff0c\u90a3\u5c31\u4ee3\u8868\u8fd8\u6ca1\u660e\u767d\u6b27\u62c9\u7b5b\u7684\u672c\u8d28\u539f\u7406\u3002<\/p>\n\n\n\n<p>\u8fd9\u4e00\u7247\u662f\u6570\u8bba\u91cc\u7d20\u6570\u7684\u5f88\u91cd\u8981\u7684\u7ae0\u8282\uff0c\u671b\u5404\u4f4d\u591a\u52a8\u624b\u52e4\u52a0\u7ec3\u4e60\u6572\u6253\uff0c\u4eca\u5929\u5c31\u5230\u6b64\u7ed3\u675f\u4e86\uff0cgood day\uff01<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u3001\u7d20\u6570\uff08\u8d28\u6570\uff09 1.\u5f15\u5165 \u7d20\u6570\u53c8\u79f0\u8d28\u6570\uff0c\u662f\u5927\u4e8e1\u7684\u81ea\u7136\u6570\u4e2d\u53ea\u80fd\u88ab1\u548c\u5b83\u672c\u8eab\u6574\u9664\u7684\u6570\u3002\u76f8\u5bf9\u7684\uff0c\u5982\u679c\u4e00\u4e2a\u5927\u4e8e1\u7684 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":156,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,15,12],"tags":[17],"class_list":["post-305","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-article","category-algorithm-template","category-programming-algorithm-road","tag-prime"],"_links":{"self":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/305","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/comments?post=305"}],"version-history":[{"count":5,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/305\/revisions"}],"predecessor-version":[{"id":311,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/posts\/305\/revisions\/311"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/media\/156"}],"wp:attachment":[{"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/media?parent=305"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/categories?post=305"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jiangqvweihuan.cn\/index.php\/wp-json\/wp\/v2\/tags?post=305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}