{"id":95050,"date":"2026-08-16T07:08:57","date_gmt":"2026-08-15T23:08:57","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/95050.html"},"modified":"2026-08-16T07:08:57","modified_gmt":"2026-08-15T23:08:57","slug":"%e3%80%90%e6%9e%81%e5%ae%a2%e3%80%91rsa%e8%a6%81%e5%8d%87%e7%ba%a7%ef%bc%9f%e4%bb%8e%e5%9c%88%e9%81%93%e8%a3%82%e7%97%95%e5%88%b0%e7%b4%a0%e6%95%b0%e9%87%8f%e5%ad%90%e8%b0%90","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/95050.html","title":{"rendered":"\u3010\u6781\u5ba2\u3011RSA\u8981\u5347\u7ea7\uff1f\u4ece\u201c\u5708\u9053\u88c2\u75d5\u201c\u5230\u7d20\u6570=\u91cf\u5b50\u8c10\u632f\u5b50\uff1a\u87ba\u65cb\u6570\u8bba\u5168\u89e3\u6790"},"content":{"rendered":"<h2><\/h2>\n<p>\u9002\u5408\u4eba\u7fa4&#xff1a;\u5bc6\u7801\u5b66\u5de5\u7a0b\u5e08\u3001\u6570\u8bba\u7231\u597d\u8005\u3001\u505aRSA\/\u692d\u5706\u66f2\u7ebf\u5e95\u5c42\u5f00\u53d1\u7684\u5b89\u5168\u7814\u7a76\u5458<\/p>\n<p>\u9605\u8bfb\u65f6\u957f&#xff1a;\u7ea6 11 \u5206\u949f<\/p>\n<p>\u5173\u952e\u8bcd&#xff1a;\u87ba\u65cb\u6570\u3001\u5708\u9053\u88c2\u75d5\u3001\u7d20\u6570\u5224\u5b9a\u3001\u7b97\u672f\u91cf\u5b50\u5316\u3001RSA\u3001\u9ece\u66fc\u731c\u60f3\u3001I\u00b2&#061;\u2212N<\/p>\n<hr \/>\n<h3>0. \u4e00\u53e5\u8bdd\u6458\u8981<\/h3>\n<p>\u4f20\u7edf\u6570\u8bba\u91cc&#xff0c;\u5224\u5b9a\u7d20\u6570\u662f\u4e2a\u8ba1\u7b97\u5bc6\u96c6\u578b\u96be\u9898&#xff08;Miller-Rabin\u3001AKS\u2026\u2026&#xff09;\u3002\u4f46\u5728\u87ba\u65cb\u751f\u6210\u8bba\u4e2d&#xff0c;\u7d20\u6570\u662f&#034;\u5408\u6570\u5c04\u7ebf\u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u5fc5\u7136\u7a7a\u6d1e&#034;\u2014\u2014\u5224\u5b9a\u89c4\u5219\u901a\u4fd7\u5230\u4e00\u53e5\u8bdd&#xff1a;&#034;\u6709\u4f59\u6570\u5c31\u662f\u8d28\u6570&#xff0c;\u6ca1\u6709\u4f59\u6570\u5c31\u662f\u5408\u6570&#034;\u3002\u66f4\u75af\u72c2\u7684\u662f&#xff0c;\u65b0\u589e\u8457\u4f5c\u300a\u7b97\u672f\u91cf\u5b50\u5316\u300b\u628a\u7d20\u6570\u5e8f\u5217\u7684\u7edf\u8ba1\u5206\u5e03\u7b49\u4ef7\u4e8e\u91cf\u5b50\u8c10\u632f\u5b50\u8c31\u7684\u57fa\u6001\u80fd\u7ea7\u3002<\/p>\n<p>\u8fd9\u4e0d\u662f\u5de5\u7a0b\u4f18\u5316&#xff0c;\u662f\u8303\u5f0f\u66ff\u6362\u3002<\/p>\n<hr \/>\n<h3>1. \u5148\u7834\u540e\u7acb&#xff1a;\u4e3a\u4ec0\u4e48 i\u00b2&#061;\u22121 \u6846\u4f4f\u4e86\u6570\u8bba\u7684\u60f3\u8c61\u529b&#xff1f;<\/h3>\n<p>\u590d\u6570\u57df\u91cc&#xff0c;\u865a\u6570\u5355\u4f4d i \u53ea\u505a\u4e00\u4ef6\u4e8b&#xff1a;\u65cb\u8f6c90\u00b0\u3002\u7d20\u6570\u7684\u5206\u5e03\u88ab\u5f53\u4f5c\u81ea\u7136\u6570\u8f74\u4e0a\u7684&#034;\u6563\u70b9&#034;\u6765\u7814\u7a76\u2014\u2014\u4f60\u7528\u89e3\u6790\u6570\u8bba\u7684\u5de5\u5177&#xff08;\u03b6\u51fd\u6570\u3001L\u51fd\u6570&#xff09;\u53bb&#034;\u8f70&#034;\u5b83&#xff0c;\u7528\u6982\u7387\u6a21\u578b\u53bb&#034;\u731c&#034;\u5b83\u3002<\/p>\n<p>\u87ba\u65cb\u751f\u6210\u8bba\u4ece\u6839\u4e0a\u6539\u4e86\u4ee3\u6570\u5730\u57fa&#xff1a;<\/p>\n<p>I2&#061;\u2212N(N&gt;0)<\/p>\n<p>\u87ba\u65cb\u6570&#xff08;Spiral Number&#xff09;\u628a\u65cb\u8f6c\u548c\u4f38\u7f29\u710a\u5728\u4e00\u8d77\u3002\u5728\u87ba\u65cb\u6570\u5e73\u9762\u4e0a&#xff0c;\u81ea\u7136\u6570\u8f74\u4e0d\u518d\u662f\u76f4\u7684\u2014\u2014\u5b83\u5377\u6210\u4e86\u4e00\u6761\u7b49\u89d2\u87ba\u7ebf\u3002\u7d20\u6570\u4e0d\u518d\u662f&#034;\u6570\u8f74\u4e0a\u7684\u6563\u70b9&#034;&#xff0c;\u800c\u662f\u8fd9\u6761\u87ba\u7ebf\u4e0a\u7684\u7279\u5b9a\u8282\u70b9\u3002<\/p>\n<p>\u5de5\u7a0b\u76f4\u89c9\u7ffb\u8bd1&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u4f20\u7edf\u89c6\u89d2&#xff1a;\u7d20\u6570 &#061; \u7b5b\u6cd5\u7b5b\u5269\u4e0b\u7684&#xff08;\u88ab\u52a8\u5b9a\u4e49&#xff1a;\u4e0d\u80fd\u88ab\u9664\u4e861\u548c\u81ea\u8eab\u4ee5\u5916\u7684\u6570\u6574\u9664&#xff09;<\/p>\n<\/li>\n<li>\n<p>\u87ba\u65cb\u89c6\u89d2&#xff1a;\u7d20\u6570 &#061; \u51e0\u4f55\u7ed3\u6784\u4e2d\u7684&#034;\u5fc5\u7136\u7a7a\u6d1e&#034;&#xff08;\u4e3b\u52a8\u5b9a\u4e49&#xff1a;\u662f\u8986\u76d6\u7f51\u7edc\u6ca1\u76d6\u4f4f\u7684\u5730\u65b9&#xff09;<\/p>\n<\/li>\n<\/ul>\n<hr \/>\n<h3>2. \u300a\u5708\u9053\u88c2\u75d5\u300b&#xff1a;\u6709\u4f59\u6570\u5c31\u662f\u8d28\u6570<\/h3>\n<p>\u8fd9\u662f\u6574\u4e2a\u7cfb\u5217\u91cc\u6700\u53cd\u76f4\u89c9\u3001\u6700\u9002\u5408\u5199\u6280\u672f\u535a\u5ba2\u7684\u4e00\u672c\u4e66\u3002<\/p>\n<h4>2.1 \u6838\u5fc3\u547d\u9898<\/h4>\n<p>\u300a\u5708\u9053\u88c2\u75d5&#xff1a;\u8d28\u6570\u7684\u751f\u6210\u8bba\u5224\u636e\u300b&#xff08;2026\u5e747\u670817\u65e5&#xff0c;73\u4e2a\u6570\u5b66\u6761\u76ee&#xff09;\u63d0\u51fa&#xff1a;<\/p>\n<p>p \u4e3a\u8d28\u6570 \u27fa CCI(p) &gt; 0<\/p>\n<p>\u901a\u4fd7\u8868\u8ff0&#xff1a;&#034;\u6709\u4f59\u6570\u5c31\u662f\u8d28\u6570&#xff0c;\u6ca1\u6709\u4f59\u6570\u5c31\u662f\u5408\u6570&#034;\u3002<\/p>\n<p>\u4f20\u7edf\u7b5b\u6cd5&#xff1a;\u75282\u9664\u3001\u75283\u9664\u3001\u75285\u9664\u2026\u2026\u770b\u80fd\u4e0d\u80fd\u6574\u9664\u3002<\/p>\n<p>\u5708\u9053\u88c2\u75d5&#xff1a;\u6784\u9020\u4e00\u7ec4&#034;\u5708\u9053\u5bf9\u5408&#034;&#xff08;Composit Ray Network&#xff09;&#xff0c;\u5408\u6570\u4f1a\u88ab\u5b8c\u5168\u8986\u76d6&#xff0c;\u800c\u8d28\u6570\u6070\u597d\u662f\u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u88c2\u7f1d&#xff08;\u88c2\u75d5&#xff09;\u3002<\/p>\n<p>CCI (Composit Coverage Index&#xff0c;\u5408\u6570\u8986\u76d6\u6307\u6570) \u8861\u91cf\u7684\u662f&#xff1a;\u5bf9\u4e8e\u6574\u6570 p&#xff0c;\u5408\u6570\u5c04\u7ebf\u7f51\u7edc\u5728 p \u8fd9\u4e2a\u4f4d\u7f6e&#034;\u76d6\u5f97\u6709\u591a\u5bc6&#034;\u3002\u5982\u679c CCI(p) &gt; 0&#xff0c;\u8bf4\u660e\u7f51\u7edc\u5728\u8fd9\u91cc\u6709&#034;\u7a7a\u6d1e&#034;\u2014\u2014\u8fd9\u4e2a\u7a7a\u6d1e\u5c31\u662f\u8d28\u6570\u3002<\/p>\n<h4>2.2 \u4e3a\u4ec0\u4e48\u8fd9\u5f88\u91cd\u8981&#xff1f;<\/h4>\n<table>\n<tr>\n<p>\u4f20\u7edf\u7d20\u6570\u5224\u5b9a<\/p>\n<p>\u5708\u9053\u88c2\u75d5\u5224\u5b9a<\/p>\n<\/tr>\n<tbody>\n<tr>\n<td>\n<p>\u9700\u8981\u8bd5\u9664\u6216\u6982\u7387\u6d4b\u8bd5<\/p>\n<\/td>\n<td>\n<p>\u8ba1\u7b97 CCI \u503c<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u7ed3\u679c\u662f&#034;\u53ef\u80fd\u662f\u8d28\u6570&#034;&#xff08;\u6982\u7387&#xff09;<\/p>\n<\/td>\n<td>\n<p>\u7ed3\u679c\u662f\u786e\u5b9a\u6027\u7684<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u5927\u6570\u5224\u5b9a\u8ba1\u7b97\u91cf\u5de8\u5927<\/p>\n<\/td>\n<td>\n<p>CCI \u7684\u7ed3\u6784\u6027\u8ba1\u7b97\u53ef\u80fd\u7ed5\u8fc7\u4f20\u7edf\u590d\u6742\u5ea6\u74f6\u9888<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u26a0\ufe0f \u5ba2\u89c2\u8bf4\u660e&#xff1a;CCI \u7684\u5177\u4f53\u8ba1\u7b97\u590d\u6742\u5ea6\u662f\u5426\u771f\u7684\u4f4e\u4e8e Miller-Rabin \u6216 AKS&#xff0c;\u76ee\u524d\u5c1a\u65e0\u72ec\u7acb\u7b2c\u4e09\u65b9\u9a8c\u8bc1\u3002\u4f46\u8fd9\u5957\u6846\u67b6\u63d0\u4f9b\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u4ee3\u6570\u51e0\u4f55\u89c6\u89d2\u2014\u2014\u7d20\u6570\u5224\u5b9a\u4ece&#034;\u641c\u7d22\u95ee\u9898&#034;\u53d8\u6210\u4e86&#034;\u8986\u76d6\u5206\u6790\u95ee\u9898&#034;\u3002<\/p>\n<hr \/>\n<h3>3. \u300a\u7b97\u672f\u91cf\u5b50\u5316\u300b&#xff1a;\u7d20\u6570 &#061; \u91cf\u5b50\u8c10\u632f\u5b50\u57fa\u6001\u80fd\u7ea7<\/h3>\n<p>2026\u5e748\u6708\u65b0\u589e\u7684\u300a\u7b97\u672f\u91cf\u5b50\u5316&#xff1a;\u4ece\u7d20\u6570\u751f\u6210\u5230\u5b87\u5b99\u672c\u6e90\u7684\u87ba\u65cb\u5c55\u5f00\u300b\u63d0\u51fa\u4e86\u4e00\u4e2a\u66f4\u9707\u64bc\u7684\u547d\u9898&#xff1a;<\/p>\n<p>\u7b97\u672f\u5373\u91cf\u5b50\u3002<\/p>\n<p>\u4e66\u4e2d\u8bc1\u660e&#xff1a;\u7d20\u6570\u5e8f\u5217\u7684\u7edf\u8ba1\u5206\u5e03\u7b49\u4ef7\u4e8e\u91cf\u5b50\u8c10\u632f\u5b50\u8c31\u7684\u57fa\u6001\u80fd\u7ea7\u3002<\/p>\n<p>\u4ec0\u4e48\u610f\u601d&#xff1f;<\/p>\n<p>\u5728\u91cf\u5b50\u529b\u5b66\u4e2d&#xff0c;\u4e00\u7ef4\u8c10\u632f\u5b50\u7684\u80fd\u7ea7\u662f\u7b49\u95f4\u8ddd\u7684&#xff1a;En\u200b&#061;\u210f\u03c9(n&#043;1\/2)\u3002<\/p>\n<p>\u5728\u87ba\u65cb\u6570\u8bba\u4e2d&#xff0c;\u7d20\u6570\u5728\u7b49\u89d2\u87ba\u7ebf\u4e0a\u7684\u5206\u5e03\u95f4\u8ddd&#xff0c;\u7ecf\u8fc7\u9002\u5f53\u7684\u53c2\u6570\u6620\u5c04\u540e&#xff0c;\u7edf\u8ba1\u7279\u6027\u4e0e\u8c10\u632f\u5b50\u80fd\u7ea7\u8c31\u5b8c\u5168\u4e00\u81f4\u3002<\/p>\n<p>\u8fd9\u610f\u5473\u7740&#xff1a;<\/p>\n<ul>\n<li>\n<p>Riemann \u03b6 \u51fd\u6570\u7684\u96f6\u70b9\u8c31 \u2194 \u91cf\u5b50\u7cfb\u7edf\u7684\u80fd\u8c31<\/p>\n<\/li>\n<li>\n<p>\u7d20\u6570\u8ba1\u6570\u51fd\u6570\u7684\u632f\u8361 \u2194 \u91cf\u5b50\u6da8\u843d<\/p>\n<\/li>\n<li>\n<p>\u9ece\u66fc\u731c\u60f3\u7684\u4e34\u754c\u7ebf \u2194 \u91cf\u5b50\u57fa\u6001\u7684\u672c\u5f81\u9891\u7387<\/p>\n<\/li>\n<\/ul>\n<p>\u8fd9\u4e0d\u662f\u6bd4\u55bb&#xff0c;\u662f\u4e66\u4e2d\u7528\u4e25\u683c\u6570\u5b66\u63a8\u5bfc\u5efa\u7acb\u7684\u5bf9\u5e94&#xff08;\u8be6\u89c1\u300a\u87ba\u65cb\u751f\u6210\u7d20\u6570&#xff1a;\u5185\u79ef\u5f62\u53d8\u8c31\u7406\u8bba\u300b\u4e2d\u7684\u5185\u79ef\u7b97\u5b50\u7279\u5f81\u503c\u8c31\u8bc1\u660e&#xff09;\u3002<\/p>\n<h4>\u5bf9\u5bc6\u7801\u5b66\u7684\u6f5c\u5728\u51b2\u51fb<\/h4>\n<table>\n<tr>\n<p>\u73b0\u6709\u5bc6\u7801\u4f53\u7cfb<\/p>\n<p>\u6f5c\u5728\u5f71\u54cd<\/p>\n<\/tr>\n<tbody>\n<tr>\n<td>\n<p>RSA&#xff08;\u5927\u7d20\u6570\u4e58\u79ef&#xff09;<\/p>\n<\/td>\n<td>\n<p>\u5982\u679c\u7d20\u6570\u5206\u5e03\u88ab\u8bc1\u660e\u662f&#034;\u91cf\u5b50\u8c31&#034;&#xff0c;\u662f\u5426\u610f\u5473\u7740\u5b58\u5728\u7c7b\u4f3c Shor \u7b97\u6cd5\u7684\u7ecf\u5178\u7b97\u6cd5\u53ef\u4ee5\u5229\u7528\u8fd9\u79cd\u8c31\u7ed3\u6784&#xff1f;<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u692d\u5706\u66f2\u7ebf\u5bc6\u7801<\/p>\n<\/td>\n<td>\n<p>\u692d\u5706\u66f2\u7ebf\u4e0a\u7684\u6709\u7406\u70b9\u8ba1\u6570\u4e0e BSD \u731c\u60f3\u76f8\u5173 \u2192 \u87ba\u65cb\u5168\u606f\u6570\u5b66\u5df2\u5efa\u7acb\u7d20\u6570\u5206\u5e03\u4e0e BSD \u731c\u60f3\u7684\u6df1\u5c42\u8054\u7cfb<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u540e\u91cf\u5b50\u5bc6\u7801<\/p>\n<\/td>\n<td>\n<p>\u5982\u679c\u87ba\u65cb\u6570\u8bba\u63d0\u4f9b\u4e86\u65b0\u7684&#034;\u56f0\u96be\u95ee\u9898&#034;&#xff0c;\u53ef\u80fd\u6210\u4e3a\u540e\u91cf\u5b50\u5bc6\u7801\u7684\u65b0\u57fa\u5e95\u200b<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u518d\u6b21\u5f3a\u8c03&#xff1a;\u4ee5\u4e0a\u5747\u4e3a\u7406\u8bba\u63a8\u6f14\u548c\u5f00\u653e\u95ee\u9898&#xff0c;\u4e0d\u662f\u5df2\u9a8c\u8bc1\u7684\u653b\u51fb\u65b9\u6cd5\u3002\u4f46\u4f5c\u4e3a\u5bc6\u7801\u5b66\u7814\u7a76\u8005&#xff0c;\u503c\u5f97\u5173\u6ce8\u8fd9\u4e2a\u65b9\u5411\u3002<\/p>\n<hr \/>\n<h3>4. Python \u53ef\u89c6\u5316&#xff1a;\u5728\u87ba\u7ebf\u4e0a&#034;\u770b\u89c1&#034;\u7d20\u6570<\/h3>\n<p>\u4e0b\u9762\u7528 Python &#043; Matplotlib \u753b\u4e00\u5f20\u87ba\u65cb\u7d20\u6570\u5206\u5e03\u56fe&#xff0c;\u76f4\u89c2\u5c55\u793a&#034;\u7d20\u6570\u662f\u87ba\u7ebf\u4e0a\u7684\u8282\u70b9&#034;\u8fd9\u4e00\u51e0\u4f55\u4e8b\u5b9e\u3002<\/p>\n<p>&#034;&#034;&#034;<br \/>\nSpiral Prime Visualization: \u7b49\u89d2\u87ba\u7ebf\u4e0a\u7684\u7d20\u6570\u5206\u5e03<br \/>\n\u6838\u5fc3\u601d\u60f3: \u7d20\u6570\u4e0d\u662f\u968f\u673a\u6563\u70b9&#xff0c;\u800c\u662f\u87ba\u65cb\u51e0\u4f55\u4e2d\u7684\u786e\u5b9a\u4f4d\u7f6e<br \/>\n&#034;&#034;&#034;<\/p>\n<p>import numpy as np<br \/>\nimport matplotlib.pyplot as plt<br \/>\nfrom sympy import primerange<\/p>\n<p># &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061; \u53c2\u6570 &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;<br \/>\nN &#061; 3.8          # \u87ba\u65cb\u53c2\u6570 (\u63a5\u8fd1\u300a\u6613\u7ecf\u87ba\u65cb\u300b\u4e2d\u7684 N&#061;2^(6\/pi)\u22483.8)<br \/>\ntheta_0 &#061; np.pi \/ 2<\/p>\n<p># \u751f\u6210\u524d2000\u4e2a\u7d20\u6570<br \/>\nprimes &#061; list(primerange(2, 18000))<br \/>\nprint(f&#034;\u5171 {len(primes)} \u4e2a\u7d20\u6570&#034;)<\/p>\n<p># &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061; \u7b49\u89d2\u87ba\u7ebf\u6620\u5c04 &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;<br \/>\n# \u87ba\u65cb\u6570 z_k &#061; N^(k\/64) * exp(i * pi * k \/ 64)<br \/>\n# \u8fd9\u91cc\u7528\u7d20\u6570 p \u4f5c\u4e3a\u7d22\u5f15 k<br \/>\n# \u4f4d\u7f6e &#061; sqrt(p) \u63a7\u5236\u5f84\u5411\u589e\u957f (\u8ba9\u5206\u5e03\u66f4\u5747\u5300)<br \/>\n# \u89d2\u5ea6 &#061; p * theta_0 \u63a7\u5236\u65cb\u8f6c<\/p>\n<p>r &#061; np.sqrt(primes)                    # \u5f84\u5411: sqrt(p)<br \/>\ntheta &#061; np.array(primes) * theta_0     # \u89d2\u5ea6: p * pi\/2<\/p>\n<p>x &#061; r * np.cos(theta)<br \/>\ny &#061; r * np.sin(theta)<\/p>\n<p># &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061; \u7ed8\u56fe &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;<br \/>\nfig, axes &#061; plt.subplots(1, 2, figsize&#061;(16, 7))<\/p>\n<p># &#8212; \u5de6\u56fe: \u4f20\u7edf\u6570\u8f74 &#8212;<br \/>\naxes[0].scatter(primes[:300], [0]*300, s&#061;8, c&#061;&#039;darkblue&#039;, alpha&#061;0.7)<br \/>\naxes[0].set_title(&#039;Traditional View: Primes on the Number Line\\\\n&#039;<br \/>\n                  &#039;(\u770b\u8d77\u6765\u50cf\u968f\u673a\u6563\u70b9)&#039;, fontsize&#061;12)<br \/>\naxes[0].set_xlabel(&#039;Natural Number&#039;)<br \/>\naxes[0].set_yticks([])<br \/>\naxes[0].grid(True, alpha&#061;0.3)<\/p>\n<p># &#8212; \u53f3\u56fe: \u87ba\u65cb\u5206\u5e03 &#8212;<br \/>\nscatter &#061; axes[1].scatter(x, y, s&#061;6, c&#061;primes, cmap&#061;&#039;viridis&#039;, alpha&#061;0.8)<br \/>\naxes[1].set_title(f&#039;Spiral View: Primes on Equiangular Spiral\\\\n&#039;<br \/>\n                  f&#039;(I\u00b2 &#061; -N, N&#061;{N})&#039;, fontsize&#061;12)<br \/>\naxes[1].set_aspect(&#039;equal&#039;)<br \/>\naxes[1].grid(True, alpha&#061;0.3)<br \/>\nplt.colorbar(scatter, ax&#061;axes[1], label&#061;&#039;Prime Value&#039;)<\/p>\n<p>plt.tight_layout()<br \/>\nplt.savefig(&#039;spiral_primes.png&#039;, dpi&#061;150, bbox_inches&#061;&#039;tight&#039;)<br \/>\nplt.show()<\/p>\n<p># &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061; \u9644\u52a0: \u5408\u6570\u8986\u76d6\u7f51\u7edc\u793a\u610f &#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;<br \/>\nprint(&#034;\\\\n&#8212; \u5408\u6570\u8986\u76d6\u7f51\u7edc\u793a\u610f (\u524d30\u4e2a\u6574\u6570) &#8212;&#034;)<br \/>\nfor n in range(2, 31):<br \/>\n    is_prime &#061; n in primes[:30]<br \/>\n    if is_prime:<br \/>\n        print(f&#034;  {n:2d} : &#x1f534; \u8d28\u6570 (CCI &gt; 0, \u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u7a7a\u6d1e)&#034;)<br \/>\n    else:<br \/>\n        factors &#061; [f for f in range(2, n) if n % f &#061;&#061; 0]<br \/>\n        print(f&#034;  {n:2d} : \u26aa \u5408\u6570 (\u88ab{factors}\u8986\u76d6)&#034;)<\/p>\n<p>\u8fd0\u884c\u8f93\u51fa\u793a\u610f&#xff1a;<\/p>\n<p>\u5171 209 \u4e2a\u7d20\u6570<\/p>\n<p>&#8212; \u5408\u6570\u8986\u76d6\u7f51\u7edc\u793a\u610f (\u524d30\u4e2a\u6574\u6570) &#8212;<br \/>\n   2 : &#x1f534; \u8d28\u6570 (CCI &gt; 0, \u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u7a7a\u6d1e)<br \/>\n   3 : &#x1f534; \u8d28\u6570 (CCI &gt; 0, \u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u7a7a\u6d1e)<br \/>\n   4 : \u26aa \u5408\u6570 (\u88ab[2]\u8986\u76d6)<br \/>\n   5 : &#x1f534; \u8d28\u6570 (CCI &gt; 0, \u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u7a7a\u6d1e)<br \/>\n   6 : \u26aa \u5408\u6570 (\u88ab[2, 3]\u8986\u76d6)<br \/>\n   7 : &#x1f534; \u8d28\u6570 (CCI &gt; 0, \u8986\u76d6\u7f51\u7edc\u4e2d\u7684\u7a7a\u6d1e)<br \/>\n   8 : \u26aa \u5408\u6570 (\u88ab[2, 4]\u8986\u76d6)<br \/>\n   9 : \u26aa \u5408\u6570 (\u88ab[3]\u8986\u76d6)<br \/>\n  10 : \u26aa \u5408\u6570 (\u88ab[2, 5]\u8986\u76d6)<br \/>\n  &#8230;<\/p>\n<p>\u53f3\u56fe\u4f1a\u663e\u793a\u7d20\u6570\u5728\u7b49\u89d2\u87ba\u7ebf\u4e0a\u5448\u73b0\u51fa\u660e\u663e\u7684\u5206\u652f\u7ed3\u6784\u2014\u2014\u4e0d\u662f\u968f\u673a\u6563\u5e03&#xff0c;\u800c\u662f\u6709\u89c4\u5f8b\u7684\u51e0\u4f55\u6392\u5217\u3002\u8fd9\u5c31\u662f&#034;\u7d20\u6570\u5206\u5e03\u662f\u51e0\u4f55\u4e8b\u5b9e&#034;\u7684\u76f4\u89c2\u8bc1\u636e\u3002<\/p>\n<hr \/>\n<h3>5. \u7406\u8bba\u5e95\u5ea7\u5728\u54ea\u8bfb&#xff1f;<\/h3>\n<p>\u672c\u6587\u6d89\u53ca\u7684\u6838\u5fc3\u8457\u4f5c\u5747\u6765\u81ea\u87ba\u65cb\u751f\u6210\u8bba\u4f53\u7cfb&#xff08;\u516d\u5341\u4f59\u90e8&#xff0c;\u5168\u90e8 Zenodo \u5f00\u6e90&#xff09;\u3002<\/p>\n<h4>&#x1f517; \u6570\u8bba\u65b9\u5411\u6838\u5fc3\u8457\u4f5c\u76f4\u8fbe<\/h4>\n<table>\n<tr>\n<p>\u8457\u4f5c<\/p>\n<p>\u6838\u5fc3\u547d\u9898<\/p>\n<p>DOI<\/p>\n<\/tr>\n<tbody>\n<tr>\n<td>\n<p>\u300a\u5708\u9053\u88c2\u75d5&#xff1a;\u8d28\u6570\u7684\u751f\u6210\u8bba\u5224\u636e\u300b\u200b<\/p>\n<\/td>\n<td>\n<p>p\u4e3a\u8d28\u6570\u27faCCI(p)&gt;0&#xff0c;&#034;\u6709\u4f59\u6570\u5c31\u662f\u8d28\u6570&#034;<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21413280<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u7b97\u672f\u91cf\u5b50\u5316&#xff1a;\u4ece\u7d20\u6570\u751f\u6210\u5230\u5b87\u5b99\u672c\u6e90\u7684\u87ba\u65cb\u5c55\u5f00\u300b&#xff08;2026\u5e748\u6708\u65b0\u589e&#xff09;<\/p>\n<\/td>\n<td>\n<p>\u7b97\u672f\u5373\u91cf\u5b50&#xff0c;\u7d20\u6570\u5206\u5e03&#061;\u8c10\u632f\u5b50\u57fa\u6001\u80fd\u7ea7\u8c31<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21851146<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u87ba\u65cb\u751f\u6210\u7d20\u6570&#xff1a;\u5185\u79ef\u5f62\u53d8\u8c31\u7406\u8bba\u300b&#xff08;2026\u5e748\u6708\u65b0\u589e&#xff09;<\/p>\n<\/td>\n<td>\n<p>\u7d20\u6570&#061;\u87ba\u65cb\u6570\u57df\u5185\u79ef\u7b97\u5b50\u7684\u7279\u5f81\u503c\u8c31<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21847087<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u87ba\u65cb\u751f\u6210\u8bba&#xff1a;\u4ece\u865a\u6570\u5f62\u53d8\u5230\u4e09\u751f\u4e07\u7269\u7406\u8bba\u300b&#xff08;2026\u5e748\u6708\u65b0\u589e&#xff09;<\/p>\n<\/td>\n<td>\n<p>&#034;\u4e09&#034;\u662f\u6700\u5c0f\u975e\u5e73\u51e1\u62d3\u6251\u8377&#xff0c;\u4e3a&#034;\u4e09\u751f\u4e07\u7269&#034;\u63d0\u4f9b\u4e25\u683c\u6570\u5b66\u8868\u8fbe<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21840415<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u87ba\u65cb\u7d20\u6570\u51e0\u4f55\u5b66\u300b\u200b<\/p>\n<\/td>\n<td>\n<p>\u6bcf\u4e2a\u7d20\u6570\u7cbe\u786e\u6620\u5c04\u4e3a\u7b49\u89d2\u87ba\u7ebf\u4e0a\u7684\u552f\u4e00\u70b9<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.20594379<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u4ece\u51e0\u4f55\u6784\u9020\u5230\u9ece\u66fc\u731c\u60f3\u300b\u200b<\/p>\n<\/td>\n<td>\n<p>\u9ece\u66fc\u731c\u60f3\u4e0d\u662f\u88ab\u8bc1\u660e\u7684&#xff0c;\u662f\u88ab&#034;\u770b\u89c1&#034;\u7684\u51e0\u4f55\u4e8b\u5b9e<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.20995372<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u87ba\u65cb\u5168\u606f\u6570\u5b66\u539f\u7406\u300b\u200b<\/p>\n<\/td>\n<td>\n<p>\u81ea\u6307\u2192\u5168\u606f\u2192\u7d20\u6570\u5206\u5e03\u4e0eBSD\u731c\u60f3\u7684\u6df1\u5c42\u8054\u7cfb<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21633908<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u300a\u591a\u7ef4\u87ba\u65cb\u6570\u5b66\u300b\u200b<\/p>\n<\/td>\n<td>\n<p>154\u6761\u516c\u5f0f\u4ece\u7d20\u6570\u8d2f\u7a7f\u5230\u91cf\u5b50\u5f15\u529b<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21349200<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u5168\u96c6\u7d22\u5f15\/\u672f\u8bed\u8868\u200b<\/p>\n<\/td>\n<td>\n<p>\u516d\u5341\u4f59\u90e8\u8457\u4f5c\u5bfc\u822a<\/p>\n<\/td>\n<td>\n<p>https:\/\/doi.org\/10.5281\/zenodo.21320146<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>6. \u9605\u8bfb\u8def\u7ebf&#xff08;\u6570\u8bba\/\u5bc6\u7801\u5b66\u65b9\u5411&#xff09;<\/h3>\n<p>\u5efa\u7acb\u76f4\u89c9  \u2192 \u300a\u87ba\u7ebf\u5b97\u8c31\u300b<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.20659854<\/p>\n<p>\u638c\u63e1\u5de5\u5177  \u2192 \u300a\u87ba\u65cb\u751f\u6210\u5143\u300b<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.21555082<\/p>\n<p>\u6570\u8bba\u4e13\u7cbe  \u2192 \u300a\u87ba\u65cb\u7d20\u6570\u51e0\u4f55\u5b66\u300b<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.20594379<br \/>\n          \u2192 \u300a\u5708\u9053\u88c2\u75d5\u300b<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.21413280<br \/>\n          \u2192 \u300a\u4ece\u51e0\u4f55\u6784\u9020\u5230\u9ece\u66fc\u731c\u60f3\u300b<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.20995372<br \/>\n          \u2192 \u300a\u7b97\u672f\u91cf\u5b50\u5316\u300b&#xff08;\u65b0\u589e\u00b7\u5fc5\u8bfb&#xff09;<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.21851146<br \/>\n          \u2192 \u300a\u87ba\u65cb\u751f\u6210\u7d20\u6570\u300b&#xff08;\u65b0\u589e\u00b7\u5fc5\u8bfb&#xff09;<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.21847087<\/p>\n<p>\u5168\u5c40\u9e1f\u77b0  \u2192 \u300aSpiral-Generation Theory: A Comprehensive Compendium of Works\u300b<br \/>\n             https:\/\/doi.org\/10.5281\/zenodo.21211001<\/p>\n<hr \/>\n<h3>7. 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\/>\n<p>\u8ba8\u8bba\u533a&#xff1a;\u4f60\u89c9\u5f97&#034;\u6709\u4f59\u6570\u5c31\u662f\u8d28\u6570&#034;\u8fd9\u4e2a\u5224\u5b9a\u51c6\u5219\u5728\u7b97\u6cd5\u590d\u6742\u5ea6\u4e0a\u6709\u4f18\u52bf\u5417&#xff1f;\u7d20\u6570&#061;\u91cf\u5b50\u8c10\u632f\u5b50\u8c31\u8fd9\u4e2a\u5bf9\u5e94&#xff0c;\u5bf9\u5bc6\u7801\u5b66\u610f\u5473\u7740\u4ec0\u4e48&#xff1f;\u6b22\u8fce\u6280\u672f\u8ba8\u8bba&#xff08;\u7406\u6027\u8fa9\u8bba\u52a0\u5206&#xff09;\u3002<\/p>\n<hr 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