{"id":881,"date":"2026-10-21T09:18:13","date_gmt":"2026-10-21T09:18:13","guid":{"rendered":"https:\/\/dailykalersrot.com\/?p=881"},"modified":"2026-10-21T09:18:13","modified_gmt":"2026-10-21T09:18:13","slug":"%e0%a6%aa%e0%a7%8d%e0%a6%b0%e0%a6%bf%e0%a6%ae%e0%a6%bf%e0%a6%9f%e0%a6%bf%e0%a6%ad-%e0%a6%b0%e0%a7%81%e0%a6%9f-%e0%a6%ae%e0%a7%8c%e0%a6%b2%e0%a6%bf%e0%a6%95-%e0%a6%b8%e0%a6%82%e0%a6%96%e0%a7%8d","status":"publish","type":"post","link":"https:\/\/dailykalersrot.com\/?p=881","title":{"rendered":"\u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f: \u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u098f\u0995\u099a\u09cd\u099b\u09a4\u09cd\u09b0 \u0985\u09a7\u09bf\u09aa\u09a4\u09bf"},"content":{"rendered":"<p>\u09ae\u09a1\u09c1\u09b2\u09be\u09b0 \u098f\u09b0\u09a5\u09ae\u09c7\u099f\u09bf\u0995\u09c7 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0\u09c7\u09b0 \u09a7\u09be\u09b0\u09a3\u09be \u09ac\u09c7\u09b6 \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac\u09aa\u09c2\u09b0\u09cd\u09a3\u0964 \u09ae\u09a1\u09c1\u09b2\u09cb 7-\u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 2-\u098f\u09b0 \u09b9\u09bf\u09b8\u09be\u09ac \u09a6\u09c7\u0996\u09be \u09af\u09be\u0995\u0964 \u098f\u0996\u09be\u09a8\u09c7 2<sup>1<\/sup>= 2, 2<sup>2<\/sup>= 4 \u098f\u09ac\u0982 2<sup>3<\/sup>= 8 \u2261 1 (mod 7)\u0964 \u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7, \u09ae\u09be\u09a4\u09cd\u09b0 \u09a4\u09bf\u09a8 \u09a7\u09be\u09aa\u09c7\u0987 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 1 \u099a\u09b2\u09c7 \u0986\u09b8\u099b\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce ord<sub>7<\/sub>(2) = 3\u0964 \u098f\u0987 \u09a7\u09be\u09b0\u09a3\u09be\u09b0 \u09b8\u0999\u09cd\u0997\u09c7\u0987 \u099c\u09dc\u09bf\u09df\u09c7 \u09b0\u09df\u09c7\u099b\u09c7 \u09ab\u09be\u09b0\u09cd\u09ae\u09be\u09b0 \u09b2\u09bf\u099f\u09b2 \u09a5\u09bf\u0993\u09b0\u09c7\u09ae (Fermat&#8217;s Little Theorem)\u0964 \u098f\u0987 \u09a8\u09bf\u09df\u09ae \u0985\u09a8\u09c1\u09af\u09be\u09df\u09c0, p \u09af\u09a6\u09bf \u098f\u0995\u099f\u09bf \u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09df \u098f\u09ac\u0982 gcd(a, p) = 1 \u09b9\u09df, \u09a4\u09ac\u09c7 a<sup>p &#8211; 1<\/sup>\u2261 1 (mod p) \u09b9\u09ac\u09c7\u0964 \u098f\u09b0 \u0985\u09b0\u09cd\u09a5 \u09b9\u09b2\u09cb, \u09ae\u09a1\u09c1\u09b2\u09cb\u099f\u09bf \u09ae\u09cc\u09b2\u09bf\u0995 \u09b9\u09b2\u09c7 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u09b8\u09ae\u09cd\u09ad\u09be\u09ac\u09cd\u09af \u0985\u09b0\u09cd\u09a1\u09be\u09b0 \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 (p &#8211; 1)\u0964 \u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa, 7 \u098f\u0995\u099f\u09bf \u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u0993\u09df\u09be\u09df \u098f\u09b0 \u099c\u09a8\u09cd\u09af \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0\u09c7\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u09b8\u09c0\u09ae\u09be 7 &#8211; 1 = 6 \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7\u0964 \u09b0\u09be\u09a4\u09c1\u09b2 \u099c\u09bf\u099c\u09cd\u099e\u09c7\u09b8 \u0995\u09b0\u09b2, \u2018\u0995\u09cb\u09a8 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u0995\u09be\u099c \u0995\u09b0\u09ac\u09c7 \u09a8\u09be \u09ad\u09be\u0987\u09df\u09be?\u2019\u0964<\/p>\n<p>\u09af\u09c7 \u09b8\u0982\u0996\u09cd\u09af\u09be\u099f\u09bf \u09a4\u09be\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0\u09c7\u09b0 \u098f\u0987 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u09b8\u09c0\u09ae\u09be \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7, \u09a4\u09be\u0995\u09c7\u0987 \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f (Primitive Root) \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, g \u09af\u09a6\u09bf p \u09ae\u09a1\u09c1\u09b2\u09cb\u09a4\u09c7 \u098f\u09ae\u09a8 \u098f\u0995\u099f\u09bf \u099a\u0995\u09cd\u09b0 \u09a4\u09c8\u09b0\u09bf \u0995\u09b0\u09c7 \u09af\u09c7\u09a8 \u09a4\u09be\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0 \u09a0\u09bf\u0995 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u09b8\u09ae\u09cd\u09ad\u09be\u09ac\u09cd\u09af \u09b8\u09c0\u09ae\u09be (p &#8211; 1)-\u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09df, \u09a4\u09ac\u09c7 g-\u0995\u09c7 p-\u098f\u09b0 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u09a4\u09ac\u09c7 \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f \u09b8\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a5\u09be\u0995\u09c7 \u09a8\u09be\u0964 \u0989\u09a6\u09be\u09b9\u09b0\u09a3 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 n = 8 \u09a7\u09b0\u09be \u09af\u09be\u0995\u0964 8-\u098f\u09b0 \u099a\u09c7\u09df\u09c7 \u099b\u09cb\u099f \u0993 \u09b8\u09b9\u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0997\u09c1\u09b2\u09cb \u09b9\u09b2\u09cb 1, 3, 5 \u0993 7\u2014\u09af\u09be\u09b0 \u09ae\u09cb\u099f \u09b8\u0982\u0996\u09cd\u09af\u09be 4\u099f\u09bf\u0964 \u0985\u09df\u09b2\u09be\u09b0\u09c7\u09b0 \u099f\u09cb\u099f\u09c7\u09a8\u09cd\u099f \u09ab\u09be\u0982\u09b6\u09a8 \u0985\u09a8\u09c1\u09af\u09be\u09df\u09c0 \u03c6(8) = 4\u0964 \u09af\u09a6\u09bf 8-\u098f\u09b0 \u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f \u09a5\u09be\u0995\u09a4, \u09a4\u09ac\u09c7 \u09a4\u09be\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0 \u09b9\u0993\u09df\u09be \u0989\u099a\u09bf\u09a4 \u099b\u09bf\u09b2 4\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b9\u09bf\u09b8\u09be\u09ac \u0985\u09a8\u09c1\u09af\u09be\u09df\u09c0 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09a4\u09c1\u0987 \u09af\u09a6\u09bf \u200b3, 5, 7\u200b-\u098f\u09b0 \u09aa\u09be\u0993\u09df\u09be\u09b0 \u0995\u09b0\u09bf\u09b8, \u09a6\u09c7\u0996\u09ac\u09bf \u099a\u09ae\u09ce\u0995\u09be\u09b0 \u098f\u0995 \u0995\u09be\u09a3\u09cd\u09a1:\u2019\u0964<\/p>\n<ul>\n<li>3<sup>2<\/sup>= 9 \u2261 1 (mod 8) \u21d2 ord<sub>8<\/sub>(3) = 2<\/li>\n<li>5<sup>2<\/sup>= 25 \u2261 1 (mod 8) \u21d2 ord<sub>8<\/sub>(5) = 2<\/li>\n<li>7<sup>2<\/sup>= 49 \u2261 1 (mod 8) \u21d2 ord<sub>8<\/sub>(7) = 2<\/li>\n<\/ul>\n<p>\u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7, \u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0\u0987 4 \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u09aa\u09cc\u0981\u099b\u09be\u09df \u09a8\u09be, \u09ac\u09b0\u0982 \u0998\u09be\u09a4 2-\u09a4\u09c7\u0987 \u09b8\u09ac\u09be\u0987 1 \u09b9\u09df\u09c7 \u09af\u09be\u09df\u0964 \u09ab\u09b2\u09c7 \u0995\u09cb\u09a8\u09cb \u09aa\u09c2\u09b0\u09cd\u09a3 \u099a\u0995\u09cd\u09b0 \u0997\u09a0\u09bf\u09a4 \u09b9\u09df \u09a8\u09be \u098f\u09ac\u0982 \u09af\u09cc\u0997\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be 8-\u098f\u09b0 \u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f \u09a5\u09be\u0995\u09c7 \u09a8\u09be\u0964 \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3\u09bf\u09a4 \u09af\u09c7, \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f \u0995\u09c7\u09ac\u09b2 \u09a4\u0996\u09a8\u0987 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a5\u09be\u0995\u09c7 \u09af\u0996\u09a8 n-\u098f\u09b0 \u09ae\u09be\u09a8 2, 4, p<sup>k<\/sup>\u0985\u09a5\u09ac\u09be 2 p<sup>k<\/sup>\u09b9\u09df (\u09af\u09c7\u0996\u09be\u09a8\u09c7 p \u098f\u0995\u099f\u09bf \u09ac\u09bf\u099c\u09cb\u09dc \u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be)\u0964<\/p>\n<h3>\u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7 \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/h3>\n<p>\u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac\u09aa\u09c2\u09b0\u09cd\u09a3 \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09b8\u09cd\u09af\u09be \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09be \u09af\u09be\u09df\u0964 \u09af\u09c7\u09ae\u09a8: p \u098f\u0995\u099f\u09bf \u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09b8\u09ae\u09cd\u09ad\u09ac \u09af\u09c7 1<sup>2<\/sup>\u22c5 2<sup>2<\/sup>\u22c5 3<sup>2<\/sup>\u22ef(p &#8211; 1)<sup>2<\/sup>\u2261 1 (mod p)\u0964 \u098f\u09b0 \u0985\u0982\u09b6 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u0989\u0987\u09b2\u09b8\u09a8\u09b8 \u09a5\u09bf\u0993\u09b0\u09c7\u09ae (Wilson&#8217;s Theorem) \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09df, \u09af\u09be \u09b9\u09b2\u09cb (p &#8211; 1)! \u2261 -1 (mod p)\u0964<\/p>\n<p>\u09a7\u09b0\u09be \u09af\u09be\u0995 p \u098f\u0995\u099f\u09bf \u09ac\u09bf\u099c\u09cb\u09dc \u09ae\u09cc\u09b2\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be \u098f\u09ac\u0982 g \u09a4\u09be\u09b0 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09b0\u09bf\u09ae\u09bf\u099f\u09bf\u09ad \u09b0\u09c1\u099f\u0964 g-\u098f\u09b0 \u0985\u09b0\u09cd\u09a1\u09be\u09b0 (p &#8211; 1) \u09b9\u0993\u09df\u09be\u09df g<sup>1<\/sup>g<sup>2<\/sup>g<sup>3<\/sup>\u2026, g<sup>p &#8211; 1<\/sup>\u098f\u09b0 \u09ad\u09be\u0997\u09b6\u09c7\u09b7\u0997\u09c1\u09b2\u09cb \u0986\u09b2\u09be\u09a6\u09be \u098f\u09ac\u0982 \u0998\u09c1\u09b0\u09bf\u09df\u09c7-\u09ab\u09bf\u09b0\u09bf\u09df\u09c7 \u098f\u09b0\u09be 1, 2, 3, \u2026, p &#8211; 1 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0997\u09c1\u09b2\u09cb\u0995\u09c7\u0987 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 1 \u09a5\u09c7\u0995\u09c7 (p &#8211; 1) \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0997\u09c1\u09b2\u09cb\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 \u09ae\u09c2\u09b2\u09a4 g-\u098f\u09b0 \u09b8\u09c2\u099a\u0995 \u09ac\u09be \u09aa\u09be\u0993\u09df\u09be\u09b0\u0997\u09c1\u09b2\u09cb\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u0986\u09ae\u09b0\u09be \u09af\u09a6\u09bf \u200b1\u200b \u09a5\u09c7\u0995\u09c7 \u200bp &#8211; 1\u200b \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u09b8\u09ac \u0995\u099f\u09bf \u09b8\u0982\u0996\u09cd\u09af\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09bf, \u09a4\u09be \u0986\u09b8\u09b2\u09c7 \u200bg\u200b-\u098f\u09b0 \u09aa\u09be\u0993\u09df\u09be\u09b0\u0997\u09c1\u09b2\u09cb \u0997\u09c1\u09a3 \u0995\u09b0\u09be\u09b0 \u09b8\u09ae\u09be\u09a8!\u2019\u0964<\/p>\n<p>\u09b8\u09c2\u099a\u0995\u09c7\u09b0 \u09a8\u09bf\u09df\u09ae\u09c7 \u0997\u09c1\u09a3\u09c7\u09b0 \u09b8\u09ae\u09df \u09aa\u09be\u0993\u09df\u09be\u09b0\u0997\u09c1\u09b2\u09cb \u09af\u09cb\u0997 \u09b9\u09df<\/p>\n<ul>\n<li>(p &#8211; 1)! = 1 \u22c5 2 \u22c5 3\u22ef(p &#8211; 1) \u2261 g<sup>1<\/sup>\u22c5 g<sup>2<\/sup>\u22c5 g<sup>3<\/sup>\u22efg<sup>p-1<\/sup>(mod p)<\/li>\n<li>g<sup>1<\/sup>\u22c5 g<sup>2<\/sup>\u22c5 g<sup>3<\/sup>\u22efg<sup>p &#8211; 1<\/sup>= g<sup>1+2+3+\u22ef+(p &#8211; 1)<\/sup>= g<sup>p(p-1)\/2<\/sup><\/li>\n<\/ul>\n<p>\u098f\u0996\u09a8 \u09ab\u09be\u09b0\u09cd\u09ae\u09be\u09b0 \u09b2\u09bf\u099f\u09b2 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