[ { "id": 491117, "publisher": 651770, "username": "即未用户8806", "avatar": "https://img-avatar.eduzhixin.com/avatar/cc53a2a7327406fc2d566e349db7684d/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "打错了，6*142857=857142", "thumb_up_num": 2, "comment_num": 0, "created_at": "1777300907", "has_thumb_up": 0, "comment_list": [ ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_d63a46abad8e425e7d2b741ae3915408" }, { "id": 491133, "publisher": 673386, "username": "羽落繁星", "avatar": "https://img-avatar.eduzhixin.com/avatar/321dc3b1e417bb05c2095e95aadfdc5a/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003col\u003e\u003cli\u003e对于斐波那契数列，前一项和后一项的比趋近于0.618(黄金分割比)\u003c/li\u003e\u003cli\u003e$985^2-211^2$=917813+7891\u003c/li\u003e\u003cli\u003e$sin^2x+cos^2x=1$,而$cosh^2x-sinh^2x=1$\u003c/li\u003e\u003cli\u003e76923*3=230769\u0026nbsp; \u0026nbsp;230769*3=692307(类似142857的轮换对称)\u003c/li\u003e\u003c/ol\u003e", "thumb_up_num": 23, "comment_num": 1, "created_at": "1777302408", "has_thumb_up": 0, "comment_list": [ { "id": 763111, "publisher": 641990, "username": "XYZ小宇宙", "avatar": "https://img-avatar.eduzhixin.com/avatar/7601c7e640f2e3734834ff3be67c2938/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e第二个是何意味\u003c/p\u003e\u003cp\u003e《细节91》《细节78》\u003c/p\u003e", "thumb_up_num": 4, "did": 66723, "created_at": "1777608161", "rid": 491133, "reply_uid": 0, "reply_uname": "", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_18a17e7e5517cda9b785d58d9de5feb4" } ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_b268a1caef4b724316cd0b5eda504bca" }, { "id": 491148, "publisher": 632467, "username": "即未用户9223", "avatar": "https://img-avatar.eduzhixin.com/avatar/61a1daf9c1c49e1a653dcc28baba97d9/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e1.不管是从哪两个数开始，按照斐波那契数列的规则递推下去，最终后一项与前一项的比总会趋近于黄金分割$\\varphi$\u003c/p\u003e\u003cp\u003e2.$\\frac{\\pi}{4}=1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\cdots$\u003c/p\u003e\u003cp\u003e3.圆的面积$S=\\pi R^2,将S对R求导得到2\\pi R$，即圆的周长\u003c/p\u003e\u003cp\u003e同样的，球的体积$V=\\frac{4}{3}\\pi R^3,求导得到4\\pi R^2,即球的表面积(求导\\rightarrow降维？)$\u003c/p\u003e", "thumb_up_num": 3, "comment_num": 1, "created_at": "1777304412", "has_thumb_up": 0, "comment_list": [ { "id": 762384, "publisher": 632467, "username": "即未用户9223", "avatar": "https://img-avatar.eduzhixin.com/avatar/61a1daf9c1c49e1a653dcc28baba97d9/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e注：1.这两个数除了0,0以外任意取(可以仅有一个是0)\u003c/p\u003e\u003cp\u003e补充一个：$e=2+\\frac{1}{1+\\frac{1}{2+\\frac{2}{3+\\cdots}}}$\u003c/p\u003e", "thumb_up_num": 3, "did": 66723, "created_at": "1777306042", "rid": 491148, "reply_uid": 0, "reply_uname": "", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_271e0c611961a8159a9989c40d24eb8c" } ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_271e0c611961a8159a9989c40d24eb8c" }, { "id": 491157, "publisher": 688559, "username": "好好学习，天天向上！", "avatar": "https://img-avatar.eduzhixin.com/avatar/cfcd208495d565ef66e7dff9f98764da/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e要说数学的美，欧拉必须榜上有名\u003c/p\u003e\u003cp\u003e$e^{i\\theta}=\\cos \\theta+i\\sin\\theta$\u003c/p\u003e\u003cp\u003e牛顿-莱布尼茨也不能少\u003c/p\u003e\u003cp\u003e若 $F\u0027(x)$ 在 $[a,b]$ 可积，则 $\\displaystyle \\int_a^bF\u0027(T)\\mathrm dt=F(b)-F(a)$\u003c/p\u003e\u003cp\u003e其实高斯和斯托克斯也很优美，大半夜的先不写了\u003c/p\u003e", "thumb_up_num": 6, "comment_num": 5, "created_at": "1777305714", "has_thumb_up": 0, "comment_list": [ { "id": 762379, "publisher": 688559, "username": "好好学习，天天向上！", "avatar": "https://img-avatar.eduzhixin.com/avatar/cfcd208495d565ef66e7dff9f98764da/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e打错一个，中间的T应为t\u003c/p\u003e", "thumb_up_num": 2, "did": 66723, "created_at": "1777305737", "rid": 491157, "reply_uid": 0, "reply_uname": "", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_a70d462f42e80abc9f8c027562f1ee09" }, { "id": 762387, "publisher": 603637, "username": "致百年前的你", "avatar": "https://img-avatar.eduzhixin.com/avatar/f5b9ec59e95675f31fa50035e6aeb6a4/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e我在《可视化微分几何与形式》里面翻到了一个理论叫做牛—莱，格林，高斯，斯托克斯都是外微分基本定理（好像是这么个名）的特例的说法\u003c/p\u003e\u003cp\u003e但是现在书在学校，那一块我也还没看到\u003c/p\u003e", "thumb_up_num": 3, "did": 66723, "created_at": "1777306211", "rid": 491157, "reply_uid": 0, "reply_uname": "", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_fdb412c38d308438d8305233e3fbaa99" }, { "id": 762396, "publisher": 688559, "username": "好好学习，天天向上！", "avatar": "https://img-avatar.eduzhixin.com/avatar/cfcd208495d565ef66e7dff9f98764da/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e虽然但是，微分几何可不是什么好玩的东西（手动狗头\u003c/p\u003e", "thumb_up_num": 2, "did": 66723, "created_at": "1777306890", "rid": 491157, "reply_uid": 603637, "reply_uname": "致百年前的你", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_a70d462f42e80abc9f8c027562f1ee09" }, { "id": 762748, "publisher": 623813, "username": "幸福健康", "avatar": "https://img-avatar.eduzhixin.com/avatar/6e7fcac2614abea00f5df61fc7596153/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e微分几何不就是加了点物理竞赛解析几何吗😧\u003c/p\u003e", "thumb_up_num": 1, "did": 66723, "created_at": "1777473922", "rid": 491157, "reply_uid": 688559, "reply_uname": "好好学习，天天向上！", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_5cb81b1d00da268c98889f470838756f" }, { "id": 763866, "publisher": 449591, "username": "沉默是金", "avatar": "https://img-avatar.eduzhixin.com/avatar/abadae75833bc02a690a9fb9243f9fe6/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e其实就叫一般的Stokes公式\u003c/p\u003e", "thumb_up_num": 0, "did": 66723, "created_at": "1777796251", "rid": 491157, "reply_uid": 603637, "reply_uname": "致百年前的你", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_e0c651bd2997e3d6c110493c89d0ed98" } ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_a70d462f42e80abc9f8c027562f1ee09" }, { "id": 491169, "publisher": 632467, "username": "即未用户9223", "avatar": "https://img-avatar.eduzhixin.com/avatar/61a1daf9c1c49e1a653dcc28baba97d9/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e1.高斯绝妙定理(就是毕导讲过的那个)，任意曲面在等距变换下曲率保持不变\u003c/p\u003e\u003cp\u003e2.欧拉\"天桥\"：$e^{i\\pi}+1=0$\u003c/p\u003e", "thumb_up_num": 3, "comment_num": 0, "created_at": "1777329956", "has_thumb_up": 0, "comment_list": [ ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_271e0c611961a8159a9989c40d24eb8c" }, { "id": 491392, "publisher": 592907, "username": "UE_°§r键子", "avatar": "https://img-avatar.eduzhixin.com/avatar/c1b5474799c77632e31645d244a47b02/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "其实把一个非7倍数的数，特别是小的，除以7，你也会发现142857的循环", "thumb_up_num": 2, "comment_num": 0, "created_at": "1777471182", "has_thumb_up": 0, "comment_list": [ ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_e70273945154f4d923e3183fe58d73f2" }, { "id": 491455, "publisher": 604865, "username": "安泊猜想（黄智英等待小九限定版）", "avatar": "https://img-avatar.eduzhixin.com/avatar/bf662bdffad91c41aa5e74b66e1663aa/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e补上面评论区的内容，超过1000字符了发不了评论，只能发回复了\u003c/p\u003e\u003cp\u003e单就1/19而言，其实也不用全算出来。\u003c/p\u003e\u003cp\u003e以下仍然是推导，看结论直接到最后\u003c/p\u003e\u003cp\u003e我们刚才说明了，奇素数 $p$ 是走马灯数当且仅当 $10$ 是 $p$ 的原根。\u003c/p\u003e\u003cp\u003e这就意味着，$\\forall1\\le x\\le p-2, p\\nmid 10^x-1$。\u003c/p\u003e\u003cp\u003e考虑 $p\\neq 5$。由费马小定理，$p\\mid 10^{p-1}-1$，\u003c/p\u003e\u003cp\u003e假设 $10$ 不是 $p$ 的原根，设 $p\\mid 10^x-1$，$1\\le x\\le p-2$，\u003c/p\u003e\u003cp\u003e则由Bezout定理，存在 $m,n\\in \\Z$，使得 $m(p-1)+nx=\\gcd(p-1,x)$。\u003c/p\u003e\u003cp\u003e从而 $10^{\\gcd(p-1,x)}\\equiv (10^{p-1})^m\\cdot(10^x)^n\\equiv 1^m\\cdot 1^n\\equiv 1(\\mod p~~)$。\u003c/p\u003e\u003cp\u003e显然 $\\gcd(p-1,x)\\mid p-1$，\u003c/p\u003e\u003cp\u003e且由于 $1\\le x\\le p-2$，有 $\\gcd(p-1,x)\\le x\\le p-2\\lt p-1$。\u003c/p\u003e\u003cp\u003e从而 $\\gcd(p-1,x)\\neq p-1$。故存在 $k\\in \\N^*$ 且 $k\\ge 2$，使得 $k\\gcd(p-1,x)=p-1$。\u003c/p\u003e\u003cp\u003e此时 $k$ 存在素因数 $q$，则 $\\gcd(p-1,x)\\mid\\displaystyle\\frac{p-1}{q}$。\u003c/p\u003e\u003cp\u003e设 $\\displaystyle\\frac{p-1}{q}=l\\cdot \\gcd(p-1,x)$。\u003c/p\u003e\u003cp\u003e则 $\\displaystyle10^{\\frac{p-1}{q}}\\equiv(10^{\\gcd(p-1,x)})^l\\equiv1^l\\equiv 1(\\mod p~~)$。\u003c/p\u003e\u003cp\u003e$\\color{red}{\\LARGE{以下是结论}}$\u003c/p\u003e\u003cp\u003e故若 $p$ 不是走马灯数，则存在一个 $p-1$ 的素因子 $q$，\u003c/p\u003e\u003cp\u003e使得 $\\displaystyle 10^{\\frac{p-1}{q}}\\equiv1(\\mod p~~)$。\u003c/p\u003e\u003cp\u003e取逆否，知若所有 $p-1$ 的素因数 $q$ 都满足 $p\\nmid 10^{\\frac{p-1}{q}}-1$，则 $p$ 是走马灯数。\u003c/p\u003e\u003cp\u003e对于 $p=19$ ，由于 $18$ 的素因数仅有 $2,3$，我们只需考虑 $10^6$ 和 $10^9$ 除以19的余数。\u003c/p\u003e\u003cp\u003e$10^3\\equiv 1000\\equiv 950+38+12\\equiv 12(\\mod 19~~)$；\u003c/p\u003e\u003cp\u003e$10^6\\equiv 12^2\\equiv 144\\equiv 133+11\\equiv 11(\\mod 19~~)$；\u003c/p\u003e\u003cp\u003e$10^9\\equiv 11\\times 12\\equiv 132\\equiv18(\\mod 19~~)$。\u003c/p\u003e\u003cp\u003e从而 $10^6$ 和 $10^9$ 除以 $19$ 的余数都不为 $1$，故19是走马灯数。\u003cbr\u003e\u003c/p\u003e\u003cp\u003e当然，19本身不算特别大，因此还可以通过观察循环节得到结果。但对于更大的数，这一方法就要快捷的多。而且其还可以用于设计程序，时间复杂度应为 $O(\\sqrt{p})$（主要是分解素因数需要的时间，计算乘方的时间复杂度要小得多）\u003c/p\u003e", "thumb_up_num": 5, "comment_num": 5, "created_at": "1777515813", "has_thumb_up": 0, "comment_list": [ { "id": 762832, "publisher": 688559, "username": "好好学习，天天向上！", "avatar": "https://img-avatar.eduzhixin.com/avatar/cfcd208495d565ef66e7dff9f98764da/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e花了一个小时左右写了一个程序，对于特定的某个数，时间复杂度应该是 $O(\\sqrt p)$，若要求小于某个数的全部走马灯数，时间复杂度是 $O(\\frac{n^\\frac32}{\\log^2 n})$。\u003c/p\u003e", "thumb_up_num": 5, "did": 66723, "created_at": "1777523613", "rid": 491455, "reply_uid": 0, "reply_uname": "", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_a70d462f42e80abc9f8c027562f1ee09" }, { "id": 763119, "publisher": 403577, "username": "「Mneisa」", "avatar": "https://img-avatar.eduzhixin.com/avatar/bc210e278be1405ca446b13a02e8e56d/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e这么强?！\u003c/p\u003e", "thumb_up_num": 1, "did": 66723, "created_at": "1777610171", "rid": 491455, "reply_uid": 0, "reply_uname": "", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_4a3fdd5ef75d954cddb821ded15d6e5a" }, { "id": 763121, "publisher": 403577, "username": "「Mneisa」", "avatar": "https://img-avatar.eduzhixin.com/avatar/bc210e278be1405ca446b13a02e8e56d/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e但是恁俩是不是搞错了，空间复杂度是O，时间复杂度是⊙\u003c/p\u003e\u003cp\u003e楼主的这个时间复杂度也不是⊙(根号p)吧，是不是算错了?\u003c/p\u003e\u003cp\u003e蒟蒻不才，许久未接触时空复杂度，如有错误，还望指正\u003c/p\u003e\u003cp\u003e(有兴趣看看我出的空间复杂度的题?)\u003c/p\u003e", "thumb_up_num": 1, "did": 66723, "created_at": "1777610636", "rid": 491455, "reply_uid": 688559, "reply_uname": "好好学习，天天向上！", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_4a3fdd5ef75d954cddb821ded15d6e5a" }, { "id": 763425, "publisher": 604865, "username": "安泊猜想（黄智英等待小九限定版）", "avatar": "https://img-avatar.eduzhixin.com/avatar/bf662bdffad91c41aa5e74b66e1663aa/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e我们好像没有这个习惯，空间和时间复杂度都混用\u003c/p\u003e\u003cp\u003esqrt p是因为分解p-1质因数要这么多时间！如果只考虑快速幂当然不用这么多！\u003c/p\u003e", "thumb_up_num": 1, "did": 66723, "created_at": "1777691078", "rid": 491455, "reply_uid": 403577, "reply_uname": "「Mneisa」", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_13fc4e9ab4605e6f17c0761a22473ed0" }, { "id": 763591, "publisher": 403577, "username": "「Mneisa」", "avatar": "https://img-avatar.eduzhixin.com/avatar/bc210e278be1405ca446b13a02e8e56d/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e明白了，感谢\u003c/p\u003e", "thumb_up_num": 0, "did": 66723, "created_at": "1777729024", "rid": 491455, "reply_uid": 604865, "reply_uname": "安泊猜想（黄智英等待小九限定版）", "has_thumb_up": 0, "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_4a3fdd5ef75d954cddb821ded15d6e5a" } ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_13fc4e9ab4605e6f17c0761a22473ed0" }, { "id": 492343, "publisher": 449591, "username": "沉默是金", "avatar": "https://img-avatar.eduzhixin.com/avatar/abadae75833bc02a690a9fb9243f9fe6/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e余元公式，还有gama函数的乘积展开式\u003c/p\u003e\u003cp\u003e一元五次方程无求根公式与galois群\u003c/p\u003e", "thumb_up_num": 1, "comment_num": 0, "created_at": "1777797490", "has_thumb_up": 0, "comment_list": [ ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_e0c651bd2997e3d6c110493c89d0ed98" }, { "id": 496014, "publisher": 604865, "username": "安泊猜想（黄智英等待小九限定版）", "avatar": "https://img-avatar.eduzhixin.com/avatar/bf662bdffad91c41aa5e74b66e1663aa/s.jpg?x-oss-process=style/avatar\u0026t=202609122212", "content": "\u003cp\u003e\u003cspan class=\"simditor-mention\" href=\"/homepage/UID_fdb412c38d308438d8305233e3fbaa99\" data-mention=\"true\" contenteditable=\"false\" target=\"_blank\" data-userid=\"603637\"\u003e@致百年前的你\u003c/span\u003e \u003c/p\u003e\u003cp\u003e学到你说的那个什么定理了（\u003c/p\u003e\u003cp\u003e高数应该不会把这玩意讲太详细，考试估计也不考，但对于我一个有意向去数学系的后面可能还会听到（（（\u003c/p\u003e\u003cp\u003e\u003cimg alt=\"IMG_20260518_101828.jpg\" src=\"https://img-forum.eduzhixin.com/forum/zhixin/uploadfiles/20260518/9aee97788c8e386f9ab63a1cc476aaac7db9205f6d.jpg\" width=\"100%\" data-filename=\"IMG_20260518_101828.jpg\"\u003e\u003cbr\u003e\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e", "thumb_up_num": 3, "comment_num": 0, "created_at": "1779070872", "has_thumb_up": 0, "comment_list": [ ], "avatar_dress_up_url": "", "avatar_dress_up_name": "", "avatar_dress_up_id": 0, "user_code": "UID_13fc4e9ab4605e6f17c0761a22473ed0" } ]