{"id":20202,"date":"2016-12-11T00:02:56","date_gmt":"2016-12-10T22:02:56","guid":{"rendered":"http:\/\/www.lecturesbureau.gr\/1\/?p=20202"},"modified":"2017-07-16T07:31:12","modified_gmt":"2017-07-16T04:31:12","slug":"archimedes-cattle-problem-part-b","status":"publish","type":"post","link":"https:\/\/www.lecturesbureau.gr\/1\/archimedes-cattle-problem-part-b\/","title":{"rendered":"To \u0392\u03bf\u03b5\u03ca\u03ba\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c5 \u0391\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b7 | \u039c\u03ad\u03c1\u03bf\u03c2 \u0392’"},"content":{"rendered":"

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\u039f\u03b9 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 \u03c0\u03bf\u03c5 \u03ba\u03b1\u03c4\u03b1\u03c3\u03c4\u03c1\u03ce\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c0\u03bf\u03af\u03b7\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9:<\/p>\n

\"formula-voeiko1\"<\/a><\/p>\n

\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03bf\u03b9 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 \u03c3\u03c4\u03bf \u0392\u03bf\u03b5\u03ca\u03ba\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03c1\u03b1\u03bc\u03bc\u03b9\u03ba\u03ad\u03c2, \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03bb\u03c5\u03b8\u03bf\u03cd\u03bd \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03bc\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03bc\u03b5\u03c4\u03c1\u03bf, \u03b1\u03c2 \u03c0\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7\u03bd \u039e \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03b1\u03c0\u03b1\u03bd\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c1\u03b7\u03c4\u03ac \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03ac\u03c3\u03b9\u03b1 \u03c4\u03bf\u03c5 \u039e. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03cc\u03bc\u03c9\u03c2 \u03bf\u03b9 \u03b1\u03c0\u03b1\u03bd\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03ba\u03ad\u03c1\u03b1\u03b9\u03bf\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af, \u03bc\u03b7\u03bb\u03af\u03c4\u03b5\u03c2, \u03c4\u03bf \u039e \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf \u03c4\u03bf\u03c5 \u03b5\u03bb\u03b1\u03c7\u03af\u03c3\u03c4\u03bf\u03c5 \u03ba\u03bf\u03b9\u03bd\u03bf\u03cd \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf\u03c5 \u03cc\u03bb\u03c9\u03bd \u03c4\u03c9\u03bd \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03c4\u03ce\u03bd. \u039b\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b1\u03c5\u03c4\u03cc \u03c5\u03c0’\u03cc\u03c8\u03b9\u03bd \u03ba\u03b1\u03c4\u03b1\u03bb\u03ae\u03b3\u03bf\u03c5\u03bc\u03b5 \u03c3\u03c4\u03b9\u03c2 \u03c3\u03c7\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c0\u03bf\u03c5 \u03b4\u03cc\u03b8\u03b7\u03ba\u03b1\u03bd \u03c3\u03c4\u03b7\u03bd \u03b1\u03c1\u03c7\u03ae.<\/p>\n

\u00ab\u0395\u03c3\u03cd \u03b4\u03b5 \u03be\u03ad\u03bd\u03b5, \u03b1\u03bd \u03bc\u03bf\u03c5 \u03c0\u03b5\u03b9\u03c2 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c0\u03cc\u03c3\u03bf\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9 \u03b2\u03cc\u03b5\u03c2 \u03c4\u03bf\u03c5 \u0389\u03bb\u03b9\u03bf\u03c5, \u03c7\u03c9\u03c1\u03b9\u03c3\u03c4\u03ac \u03bc\u03b5\u03bd \u03c4\u03bf \u03c0\u03bb\u03ae\u03b8\u03bf\u03c2 \u03c4\u03c9\u03bd \u03b1\u03ba\u03bb\u03bf\u03b8\u03c1\u03b5\u03bc\u03bc\u03ad\u03bd\u03c9\u03bd \u03c4\u03b1\u03cd\u03c1\u03c9\u03bd,\u03c7\u03c9\u03c1\u03b9\u03c3\u03c4\u03ac \u03b4\u03b5 \u03c0\u03ac\u03bb\u03b9 \u03c0\u03cc\u03c3\u03b5\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9 \u03b1\u03b3\u03b5\u03bb\u03ac\u03b4\u03b5\u03c2 \u03c4\u03bf\u03c5 \u03ba\u03ac\u03b8\u03b5 \u03c7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2, \u03b4\u03b5\u03bd \u03b8\u03b1 \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03b7\u03c1\u03af\u03b6\u03b5\u03c3\u03b1\u03b9 \u03c9\u03c2 \u03b1\u03bc\u03b1\u03b8\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03b1\u03b4\u03b1\u03ae\u03c2 \u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2, \u03cc\u03bc\u03c9\u03c2 \u03b4\u03b5 \u03b8\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b1\u03c4\u03b1\u03bb\u03ad\u03b3\u03b5\u03c3\u03b1\u03b9 \u03b1\u03ba\u03cc\u03bc\u03b7 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03c4\u03c9\u03bd \u03c3\u03bf\u03c6\u03ce\u03bd. \u0391\u03bb\u03bb\u03ac \u03ad\u03bb\u03b1 \u03c3\u03ba\u03ad\u03c8\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c4\u03b9\u03c2 \u03b1\u03ba\u03cc\u03bb\u03bf\u03c5\u03b8\u03b5\u03c2 \u03c3\u03c7\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c3\u03c4\u03bf\u03c5\u03c2 \u03b2\u03cc\u03b5\u03c2 \u03c4\u03bf\u03c5 \u0389\u03bb\u03b9\u03bf\u03c5\u00bb.<\/p>\n

\u039c\u03b1\u03c2 \u03c0\u03c1\u03bf\u03b5\u03b9\u03b4\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd \u03bf \u0391\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b7\u03c2 \u03b3\u03b9\u03b1 \u03c4\u03bf \u03c4\u03b9 \u03c0\u03c1\u03cc\u03ba\u03b5\u03b9\u03c4\u03b1\u03b9 \u03bd\u03b1 \u03c3\u03c5\u03bd\u03b1\u03bd\u03c4\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5. \u0391\u03c2 \u03b4\u03bf\u03cd\u03bc\u03b5!<\/p>\n

\u00abO\u03b9 \u03bc\u03b5\u03bd \u03bb\u03b5\u03c5\u03ba\u03bf\u03af \u03c4\u03b1\u03cd\u03c1\u03bf\u03b9 \u03cc\u03c4\u03b1\u03bd \u03b1\u03bd\u03b1\u03bc\u03b9\u03c7\u03b8\u03bf\u03cd\u03bd \u03bc\u03b5 \u03c4\u03bf \u03c0\u03bb\u03ae\u03b8\u03bf\u03c2 \u03c4\u03c9\u03bd \u03ba\u03c5\u03b1\u03bd\u03ce\u03bd, \u03c3\u03c4\u03ad\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03b5 \u03c3\u03c5\u03bc\u03c0\u03b1\u03b3\u03ae \u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03cc \u03bf \u03bf\u03c0\u03bf\u03af\u03bf\u03c2 \u03ad\u03c7\u03b5\u03b9 \u03c4\u03bf \u03af\u03b4\u03b9\u03bf \u03bc\u03ad\u03c4\u03c1\u03bf \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03b2\u03ac\u03b8\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03c0\u03bb\u03ac\u03c4\u03bf\u03c2, \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03b1\u03c0\u03ad\u03c1\u03b1\u03bd\u03c4\u03b5\u03c2 \u03c0\u03b5\u03b4\u03b9\u03ac\u03b4\u03b5\u03c2 \u03c4\u03b7\u03c2 \u0398\u03c1\u03b9\u03bd\u03b1\u03ba\u03af\u03b1\u03c2 \u03b3\u03b5\u03bc\u03af\u03b6\u03bf\u03c5\u03bd \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03b1\u03c5\u03c4\u03cc \u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03cc. \u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03bf\u03b9 \u03be\u03b1\u03bd\u03b8\u03bf\u03af \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03c0\u03bf\u03b9\u03ba\u03b9\u03bb\u03cc\u03c7\u03c1\u03c9\u03bc\u03bf\u03b9 \u03b1\u03bd \u03c3\u03c5\u03bd\u03b1\u03b8\u03c1\u03bf\u03b9\u03c3\u03c4\u03bf\u03cd\u03bd \u03bc\u03b1\u03b6\u03af \u03c3\u03c4\u03ad\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03ad\u03c4\u03bf\u03b9\u03bf \u03c4\u03c1\u03cc\u03c0\u03bf \u03ce\u03c3\u03c4\u03b5, \u03b1\u03c1\u03c7\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1\u03bd, \u03bd\u03b1 \u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bd \u03c3\u03c4\u03b1\u03b4\u03b9\u03b1\u03ba\u03ac \u03c4\u03c1\u03b9\u03b3\u03c9\u03bd\u03b9\u03ba\u03cc \u03c3\u03c7\u03ae\u03bc\u03b1, \u03c7\u03c9\u03c1\u03af\u03c2 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03cc\u03bd\u03c4\u03b5\u03c2 \u03bf\u03b9 \u03c4\u03b1\u03cd\u03c1\u03bf\u03b9 \u03c4\u03c9\u03bd \u03ac\u03bb\u03bb\u03c9\u03bd \u03c7\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd, \u03bf\u03cd\u03c4\u03b5 \u03bd\u03b1 \u03bb\u03b5\u03af\u03c0\u03b5\u03b9 \u03ba\u03b1\u03bd\u03b5\u03af\u03c2\u00bb.<\/p>\n

\u0395\u03b4\u03ce \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03b4\u03cd\u03c3\u03ba\u03bf\u03bb\u03bf \u03bc\u03ad\u03c1\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2. \u0397 \u03c0\u03c1\u03ce\u03c4\u03b7 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b7 \u03bb\u03ad\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03bf \u039b+\u039a \u03af\u03c3\u03bf\u03bd \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf, \u03c0\u03bf\u03c5 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c4\u03c9\u03bd \u039b+\u039a \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5, \u03b4\u03af\u03bd\u03b5\u03b9 10366482\u03bd+7460514=m2<\/sup> \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae:<\/p>\n

22<\/sup>\u22193\u221911\u221929\u22194657\u2219\u03bd+ m2 <\/sup>\u03b1\u03c0\u03cc \u03cc\u03c0\u03bf\u03c5 \u03b1\u03c0\u03b1\u03b9\u03c4\u03b5\u03af\u03c4\u03b1\u03b9 \u03bf v \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 3\u221911\u221929\u22194657\u2219p2 <\/sup><\/p>\n

\u0393\u03b9\u03b1 \u03c4\u03b7\u03bd \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03b7 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b7 \u03c4\u03b1 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b9\u03bf \u03c0\u03b5\u03c1\u03af\u03c0\u03bb\u03bf\u03ba\u03b1 \u03ba\u03b1\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bd \u03bd\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bd \u03bc\u03b5 \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf\u03c5\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03c4\u03b7\u03c2 \u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 1+2+…n. \u03a4\u03bf\u03bd \u03b3\u03bd\u03c9\u03c3\u03c4\u03cc \u03bc\u03b1\u03c2 \u03c4\u03cd\u03c0\u03bf 1+2+…n=n(n+1)\/2 \u03bf\u03b9 \u03b1\u03c1\u03c7\u03b1\u03af\u03bf\u03b9 \u03c4\u03bf\u03bd \u03ae\u03be\u03b5\u03c1\u03b1\u03bd, \u03b1\u03c6\u03bf\u03cd \u03c0\u03b1\u03c1\u03b1\u03b4\u03b5\u03af\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c7\u03ac\u03c1\u03b9\u03bd \u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c9\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c3\u03c4\u03bf\u03bd \u039d\u03b9\u03ba\u03cc\u03bc\u03b1\u03c7\u03bf, \u03c3\u03b5 \u03ad\u03bd\u03b1 \u03bc\u03ac\u03bb\u03bb\u03bf\u03bd \u03b1\u03c0\u03bb\u03cc \u03b2\u03b9\u03b2\u03bb\u03af\u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03ae\u03c2. \u0393\u03c1\u03ac\u03c6\u03b5\u03b9 \u03bf \u039d\u03b9\u03ba\u03cc\u03bc\u03b1\u03c7\u03bf\u03c2,<\/p>\n

\u00ab\u0394\u03bf\u03b8\u03b5\u03bd\u03c4\u03c9\u03bd \u03b1\u03c0\u03cc \u03bc\u03bf\u03bd\u03ac\u03b4\u03bf\u03c2 \u03bf\u03c0\u03bf\u03c3\u03c9\u03c3\u03bf\u03cd\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03b5\u03c6\u03b5\u03be\u03ae\u03c2 \u03b5\u03c5\u03c1\u03b5\u03af\u03bd, \u03cc\u03c3\u03bf\u03c2 \u03b5\u03c3\u03c4\u03af\u03bd \u03bf \u03c3\u03cd\u03bc\u03c0\u03b1\u03c2\u00bb<\/p>\n

\u03ba\u03b1\u03b9 \u03b1\u03c0\u03b1\u03bd\u03c4\u03ac \u03bf \u03af\u03b4\u03b9\u03bf\u03c2 \u03bb\u03af\u03b3\u03bf \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9,<\/p>\n

\u00ab\u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03bc\u03b5\u03bd \u03c4\u03bf\u03bd \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03b5\u03c0\u03af \u03c4\u03bf\u03bd \u03b5\u03c6\u03b5\u03be\u03ae\u03c2 \u03b1\u03c5\u03c4\u03bf\u03cd \u03c4\u03bf\u03c5 \u03bc\u03b5\u03af\u03b6\u03bf\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03b3\u03b5\u03bd\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c4\u03bf \u03ae\u03bc\u03b9\u03c3\u03c5 \u03bb\u03b1\u03b2\u03cc\u03bd\u03c4\u03b5\u03c2 \u03ad\u03be\u03bf\u03bc\u03b5\u03bd \u03c4\u03bf\u03bd \u03c3\u03cd\u03bc\u03c0\u03b1\u03bd\u03c4\u03b1\u00bb.<\/p>\n

\u03a3\u03c5\u03bd\u03b1\u03b8\u03c1\u03bf\u03af\u03b6\u03bf\u03bd\u03c4\u03b5\u03c2 \u03cc\u03bb\u03b1 \u03c4\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9, \u03b7 \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03b7 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b7 \u03c4\u03bf\u03c5 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u039e+\u03a0=n(n+1)\/2 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 (\u03c3\u03b5 \u03c0\u03b1\u03c1\u03b1\u03b3\u03bf\u03bd\u03c4\u03bf\u03c0\u03bf\u03b9\u03b7\u03bc\u03ad\u03bd\u03b7 \u03bc\u03bf\u03c1\u03c6\u03ae)<\/p>\n

3\u22197\u221911\u221929\u2219353\u2219(4657) 2<\/sup>\u2219p2 <\/sup>n(n+1)\/2<\/p>\n

\u039c\u03b5 \u03c3\u03cd\u03b3\u03c7\u03c1\u03bf\u03bd\u03b1 \u03c3\u03cd\u03bc\u03b2\u03bf\u03bb\u03b1 \u03b8\u03b1 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03c3\u03b1\u03bc\u03b5 \u03bd\u03b1 \u03c3\u03c5\u03bd\u03b5\u03c7\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c9\u03c2 \u03b5\u03be\u03ae\u03c2. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b4\u03cd\u03bf \u03bc\u03ad\u03bb\u03b7 \u03b5\u03c0\u03af 8 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b1\u03c6\u03bf\u03cd 80n(n+1)\/2))+1=(2n+1)2<\/sup>, \u03bf\u03b4\u03b7\u03b3\u03bf\u03cd\u03bc\u03b1\u03c3\u03c4\u03b5 \u03c3\u03b5 \u03bc\u03af\u03b1 \u03c4\u03b9\u03c2 \u03b1\u03c0\u03bf\u03ba\u03b1\u03bb\u03bf\u03cd\u03bc\u03b5\u03bd\u03b5\u03c2 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 Pell:<\/p>\n

(2n+1)2<\/sup>-2\u22193\u22197\u221911\u221929\u2219353\u2219(4657\u2219p)2 <\/sup>=1<\/p>\n

\u0398\u03b1 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03c3\u03b1\u03bd \u03cc\u03bc\u03c9\u03c2 \u03bf\u03b9 \u03b1\u03c1\u03c7\u03b1\u03af\u03bf\u03b9 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03af \u03bd\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bd \u03b1\u03c5\u03c4\u03ac \u03c4\u03b1 \u03b2\u03ae\u03bc\u03b1\u03c4\u03b1; H \u03b1\u03c0\u03ac\u03bd\u03c4\u03b7\u03c3\u03b7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1\u03b9, \u03b1\u03bd \u03b4\u03b5\u03b9 \u03ba\u03b1\u03bd\u03b5\u03af\u03c2 \u03c4\u03b7\u03bd \u03bb\u03cd\u03c3\u03b7 \u03c4\u03bf\u03c5 \u0394\u03b9\u03bf\u03c6\u03ac\u03bd\u03c4\u03bf\u03c5 \u03c3\u03c4\u03bf 4, 38 \u03c4\u03c9\u03bd \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03ce\u03bd \u03c4\u03bf\u03c5, \u03cc\u03c0\u03bf\u03c5 \u03b3\u03c1\u03ac\u03c6\u03b5\u03b9:<\/p>\n

\u00ab\u03c0\u03b1\u03c2 \u03b4\u03b5 \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf\u03c2 \u03bf\u03ba\u03c4\u03ac\u03ba\u03b9\u03c2 \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03bb\u03b1\u03b2\u03ce\u03bd \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1, \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf\u03c2 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9\u00bb.<\/p>\n

\u0391\u03c5\u03c4\u03cc \u03cc\u03bc\u03c9\u03c2 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03b5\u03b9 \u03c3\u03b7\u03bc\u03b1\u03c3\u03af\u03b1 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03bd o \u0394\u03b9\u03cc\u03c6\u03b1\u03bd\u03c4\u03bf\u03c2, \u03bf\u03c0\u03bf\u03af\u03bf\u03c2 \u03ad\u03c7\u03b5\u03b9 \u03c0\u03c1\u03bf\u03c7\u03c9\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf \u03b1\u03bb\u03b3\u03b5\u03b2\u03c1\u03b9\u03ba\u03cc \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03b9\u03c3\u03bc\u03cc, \u03ae\u03be\u03b5\u03c1\u03b5 \u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1. T\u03bf \u03b5\u03c1\u03ce\u03c4\u03b7\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03b9 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c7\u03c9\u03c1\u03af\u03c2 \u03c0\u03c1\u03bf\u03c7\u03c9\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03b9\u03c3\u03bc\u03cc. \u0395\u03c5\u03c4\u03c5\u03c7\u03ce\u03c2 \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03c7\u03c9\u03c1\u03af\u03bf \u03c4\u03bf\u03c5 \u03a0\u03bb\u03bf\u03c5\u03c4\u03ac\u03c1\u03c7\u03bf\u03c5 \u03c0\u03bf\u03c5 \u03bb\u03ad\u03b5\u03b9 \u03c4\u03b1 \u03af\u03b4\u03b9\u03b1 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b1, \u03c0\u03bf\u03bb\u03cd \u03b5\u03bd\u03c9\u03c1\u03af\u03c4\u03b5\u03c1\u03b1:<\/p>\n

\u00ab\u03c0\u03b1\u03c2 \u03b3\u03ac\u03c1 \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03bf\u03ba\u03c4\u03ac\u03ba\u03b9\u03c2 \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1 \u03c0\u03c1\u03bf\u03c3\u03bb\u03b1\u03b2\u03ce\u03bd \u03b3\u03af\u03b3\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf\u03c2\u00bb.<\/p>\n

\u03a5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03bc\u03ad\u03b8\u03bf\u03b4\u03bf\u03c2 \u03bd\u03b1 \u03bb\u03cd\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03bf\u03b9 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 Pell \u03bc\u03b5 \u03c7\u03c1\u03ae\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03c0\u03b5\u03c1\u03b9\u03cc\u03b4\u03bf\u03c5 \u03c3\u03c5\u03bd\u03b5\u03c7\u03bf\u03cd\u03c2 \u03ba\u03bb\u03ac\u03c3\u03bc\u03b1\u03c4\u03bf\u03c2 (\u03b4\u03b5\u03bd \u03b8\u03b1 \u03bc\u03c0\u03bf\u03cd\u03bc\u03b5 \u03c3\u03c4\u03b9\u03c2 \u03bb\u03b5\u03c0\u03c4\u03bf\u03bc\u03ad\u03c1\u03b5\u03b9\u03b5\u03c2). \u039a\u03b1\u03c4’ \u03b1\u03c1\u03c7\u03ae\u03bd \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03ae \u03bc\u03b1\u03c2 \u03bb\u03cd\u03bd\u03b5\u03c4\u03b1\u03b9, \u03bc\u03cc\u03bd\u03bf \u03c0\u03bf\u03c5 \u03b7 \u03bc\u03ad\u03b8\u03bf\u03b4\u03bf\u03c2 \u03c0\u03b1\u03c1\u03ac\u03b3\u03b5\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03c4\u03b5\u03c1\u03b1\u03c3\u03c4\u03af\u03c9\u03bd \u03b4\u03b9\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03c9\u03bd: \u03a4\u03bf \u03c3\u03c5\u03bd\u03b5\u03c7\u03ad\u03c2 \u03ba\u03bb\u03ac\u03c3\u03bc\u03b1 \u03ad\u03c7\u03b5\u03b9 \u03c0\u03b5\u03c1\u03af\u03bf\u03b4\u03bf \u03bc\u03b5\u03c4\u03ac \u03b1\u03c0\u03cc 96 \u03b2\u03ae\u03bc\u03b1\u03c4\u03b1! \u03a4\u03b1 \u03b2\u03ae\u03bc\u03b1\u03c4\u03b1 \u03b1\u03c5\u03c4\u03ac \u03ad\u03b3\u03b9\u03bd\u03b1\u03bd \u03b3\u03b9\u03b1 \u03c0\u03c1\u03ce\u03c4\u03b7 \u03c6\u03bf\u03c1\u03ac \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd Amthor 1880 (\u03b2\u03bb\u03ad\u03c0\u03b5 Zeitsch, f\u00fcr Math. und Phys. \u03c4\u03cc\u03bc\u03bf\u03c2 25, 1880), \u03ba\u03b1\u03b9 \u03ad\u03c0\u03c1\u03b5\u03c0\u03b5 \u03bd\u03b1 \u03c0\u03b5\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf\u03c5\u03bc\u03b5 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03bf 1965 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b2\u03c1\u03b5\u03b8\u03b5\u03af \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac \u03b7 \u03b1\u03c0\u03ac\u03bd\u03c4\u03b7\u03c3\u03b7 \u03c3\u03c4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1.<\/p>\n

\u0393\u03b9\u03b1 \u03bd\u03b1 \u03ba\u03bb\u03b5\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03b1\u03c5\u03c4\u03ae \u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03ac\u03b3\u03c1\u03b1\u03c6\u03bf, \u03b1\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03c4\u03bf \u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c2 \u03cc\u03c4\u03b9 \u03c4\u03bf \u0392\u03bf\u03b5\u03ca\u03ba\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03b1\u03bd\u03ac\u03b3\u03b5\u03c4\u03b1\u03b9 \u03c3\u03b5 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 Pell, \u03b5\u03bd\u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03ac\u03c0\u03bf\u03c8\u03b7 \u03cc\u03c4\u03b9 \u03c4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03ad\u03c7\u03b5\u03b9 \u0391\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b5\u03b9\u03b1 \u03c0\u03c1\u03bf\u03ad\u03bb\u03b5\u03c5\u03c3\u03b7. \u039f \u0391\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b7\u03c2 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5 \u03ad\u03bc\u03bc\u03b5\u03c3\u03b1 \u03c4\u03b7\u03bd \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03c3\u03c4\u03b9\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03b3\u03b3\u03af\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \u03c4\u03b7\u03c2 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae\u03c2 \u03c1\u03af\u03b6\u03b1\u03c2 \u03c4\u03bf\u03c5 3, \u03b5\u03bd\u03ce \u03b1\u03c0\u03cc \u03c4\u03cc\u03c4\u03b5 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03bf\u03bd Fermat \u03c4\u03b7\u03bd 16\u03bf \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b4\u03b5\u03bd \u03be\u03b1\u03bd\u03b1\u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9.<\/p>\n

\u00ab\u0391\u03c6\u03bf\u03cd \u03b2\u03c1\u03b5\u03b9\u03c2 \u03ba\u03b1\u03b9 \u03b1\u03c5\u03c4\u03ac \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03b9\u03bb\u03ac\u03b2\u03b5\u03b9\u03c2 \u03c3\u03c4\u03bf\u03bd \u03bd\u03bf\u03c5 \u03c3\u03bf\u03c5, \u03ba\u03b1\u03b9 \u03b5\u03ba\u03c6\u03c1\u03ac\u03c3\u03b5\u03b9\u03c2 \u03cc\u03bb\u03b1 \u03c4\u03b1 \u03bc\u03b5\u03b3\u03ad\u03b8\u03b7 \u03c4\u03c9\u03bd \u03c0\u03bb\u03b7\u03b8\u03ce\u03bd, \u03be\u03ad\u03bd\u03b5, \u03c0\u03ae\u03b3\u03b1\u03b9\u03bd\u03b5 \u03c5\u03c0\u03b5\u03c1\u03b7\u03c6\u03b1\u03bd\u03b5\u03c5\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03cc\u03c4\u03b9 \u03b1\u03bd\u03b1\u03b4\u03b5\u03af\u03c7\u03b8\u03b7\u03ba\u03b5\u03c2 \u03bd\u03b9\u03ba\u03b7\u03c4\u03ae\u03c2 \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03c2 \u03cc\u03c4\u03b9 \u03ba\u03c1\u03af\u03b8\u03b7\u03ba\u03b5\u03c2 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, \u03c3’\u03b1\u03c5\u03c4\u03bf\u03cd \u03c4\u03bf\u03c5 \u03b5\u03af\u03b4\u03bf\u03c5\u03c2 \u03c4\u03b7\u03bd \u03c3\u03bf\u03c6\u03af\u03b1\u00bb<\/p>\n

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 <\/p>\n

 <\/p>\n

\u039c\u03ad\u03c1\u03bf\u03c2 \u0391’:\u00a0http:\/\/www.lecturesbureau.gr\/1\/the-archimedes-cattle-problem-part-a\/<\/a><\/p>\n

 <\/p>\n

 <\/p>\n

\u0391\u03a1\u03a7\u0391\u0399\u0391 \u0395\u039b\u039b\u0397\u039d\u0399\u039a\u0391 \u039c\u0391\u0398\u0397\u039c\u0391\u03a4\u0399\u039a\u0391 – \u039a\u0395\u0399\u039c\u0395\u039d\u0391 \u0399\u03a3\u03a4\u039f\u03a1\u0399\u0391\u03a3 \u039a\u0391\u0399 \u03a6\u0399\u039b\u039f\u03a3\u039f\u03a6\u0399\u0391\u03a3<\/strong><\/p>\n

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