{"id":27949,"date":"2018-08-15T00:02:49","date_gmt":"2018-08-14T21:02:49","guid":{"rendered":"http:\/\/www.lecturesbureau.gr\/1\/?p=27949"},"modified":"2018-08-14T17:48:02","modified_gmt":"2018-08-14T14:48:02","slug":"syracuses-problem-1448","status":"publish","type":"post","link":"https:\/\/www.lecturesbureau.gr\/1\/syracuses-problem-1448\/","title":{"rendered":"\u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd \u03a3\u03c5\u03c1\u03b1\u03ba\u03bf\u03c5\u03c3\u03ce\u03bd &#8211; \u0391\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03af\u03b1 \u03c7\u03b1\u03bb\u03b1\u03b6\u03b9\u03bf\u03cd"},"content":{"rendered":"<p><strong>\u0388\u03bd\u03b1 \u03c3\u03ba\u03bb\u03b7\u03c1\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03ba\u03b1\u03b9 \u03b7 \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03b5\u03bd\u03cc\u03c2! <\/strong><br \/>\n<em>&#8220;\u0397 \u03b1\u03c0\u03cc\u03bb\u03c5\u03c4\u03b7 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bf\u03b4\u03b7\u03b3\u03b5\u03af \u03c3\u03c4\u03b7\u03bd \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03b5\u03bd\u03cc\u03c2.&#8221;<\/em><br \/>\n\u039c\u03bf\u03bd\u03c4\u03b5\u03c3\u03ba\u03b9\u03ad<br \/>\n\u03a3\u03c4\u03b9\u03c2 6 \u0399\u03bf\u03c5\u03bb\u03af\u03bf\u03c5 1910, \u03b3\u03b5\u03bd\u03bd\u03ae\u03b8\u03b7\u03ba\u03b5 \u03bf \u0393\u03b5\u03c1\u03bc\u03b1\u03bd\u03cc\u03c2 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 Lothar Collatz. To 1937 \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b5 \u03bc\u03b9\u03b1 \u03b5\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u03c0\u03bf\u03c5 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9 \u03ac\u03bb\u03c5\u03c4\u03b7. \u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd \u03a3\u03c5\u03c1\u03b1\u03ba\u03bf\u03c5\u03c3\u03ce\u03bd. \u0388\u03bd\u03b1 \u03ba\u03b1\u03c4\u03ac \u03b2\u03ac\u03c3\u03b7 \u03b1\u03c0\u03bb\u03cc \u03c3\u03c4\u03b7\u03bd \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b7 \u03b5\u03c1\u03ce\u03c4\u03b7\u03bc\u03b1 \u03c0\u03bf\u03c5 \u03cc\u03bc\u03c9\u03c2 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03c7\u03c4\u03b7\u03ba\u03b5 \u03c0\u03bf\u03bb\u03cd \u03b4\u03cd\u03c3\u03ba\u03bf\u03bb\u03bf \u03bd\u03b1 \u03b1\u03c0\u03b1\u03bd\u03c4\u03b7\u03b8\u03b5\u03af. \u039f \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1\u03c2 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03c4\u03bf \u03c4\u03c3\u03b5\u03ba\u03ac\u03c1\u03b5\u03b9 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ce\u03bd\u03c4\u03b1\u03c2 \u03bc\u03bf\u03bb\u03cd\u03b2\u03b9 \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03c4\u03af \u03ae \u03b1\u03ba\u03cc\u03bc\u03b1 \u03ba\u03b1\u03b9 \u03ad\u03bd\u03b1 \u03b1\u03c0\u03bb\u03cc \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03c4\u03ae \u03c4\u03c3\u03ad\u03c0\u03b7\u03c2. \u0395\u03af\u03bd\u03b1\u03b9 \u03c4\u03cc\u03c3\u03bf \u03b1\u03c0\u03bb\u03cc \u03c3\u03c4\u03b7\u03bd \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b7 \u03b5\u03be\u03b1\u03b9\u03c1\u03b5\u03c4\u03b9\u03ba\u03ac \u03b4\u03cd\u03c3\u03ba\u03bf\u03bb\u03bf \u03cc\u03bc\u03c9\u03c2 \u03bd\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03b9\u03c7\u03b8\u03b5\u03af. \u03a4\u03bf \u03b5\u03af\u03b4\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c0\u03bf\u03c5 \u03bd\u03bf\u03bc\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b1\u03c0\u03ac\u03bd\u03c4\u03b7\u03c3\u03b7 \u03b1\u03bb\u03bb\u03ac \u03b4\u03b5\u03bd \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03c4\u03bf \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5.<br \/>\n\u03a3\u03ba\u03b5\u03c6\u03c4\u03b5\u03af\u03c4\u03b5 \u03ad\u03bd\u03b1\u03bd \u03b8\u03b5\u03c4\u03b9\u03ba\u03cc \u03b1\u03ba\u03ad\u03c1\u03b1\u03b9\u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc ,\u03cc\u03c0\u03bf\u03b9\u03bf\u03bd \u03b5\u03c3\u03b5\u03af\u03c2 \u03b8\u03ad\u03bb\u03b5\u03c4\u03b5.<br \/>\n\u0391\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03b5\u03af\u03c3\u03c4\u03b5 \u03c4\u03ce\u03c1\u03b1 \u03c4\u03b7\u03bd \u03b5\u03be\u03ae\u03c2 \u03b4\u03b9\u03b1\u03b4\u03b9\u03ba\u03b1\u03c3\u03af\u03b1.<\/p>\n<p>-\u0391\u03bd \u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ac\u03c1\u03c4\u03b9\u03bf\u03c2 (\u03b6\u03c5\u03b3\u03cc\u03c2),\u03b4\u03b9\u03b1\u03b9\u03c1\u03ad\u03c3\u03c4\u03b5 \u03c4\u03bf\u03bd \u03bc\u03b5 \u03c4\u03bf 2.<br \/>\n-\u0391\u03bd \u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b5\u03c1\u03b9\u03c4\u03c4\u03cc\u03c2 (\u03bc\u03bf\u03bd\u03cc\u03c2), \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03c3\u03c4\u03b5 \u03c4\u03bf\u03bd \u03bc\u03b5 \u03c4\u03bf 3 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c3\u03c4\u03b5 \u03c4\u03bf 1.<br \/>\n\u0395\u03c0\u03b1\u03bd\u03b1\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 \u03c4\u03b7\u03bd \u03b4\u03b9\u03b1\u03b4\u03b9\u03ba\u03b1\u03c3\u03af\u03b1. \u03a0\u03b1\u03c1\u03b1\u03c4\u03b7\u03c1\u03b5\u03af\u03c3\u03c4\u03b5 \u03c4\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1.<\/p>\n<p>\u0395\u03b3\u03ce \u03c3\u03ba\u03ad\u03c6\u03c4\u03b7\u03ba\u03b1 \u03c4\u03bf 11.\u0395\u03af\u03bd\u03b1\u03b9 \u03c0\u03b5\u03c1\u03b9\u03c4\u03c4\u03cc\u03c2, \u03ac\u03c1\u03b1 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03c9 \u03bc\u03b5 3 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c4\u03c9 1,\u03bf \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 3&#215;11+1=34,\u03bf 34 \u03c4\u03ce\u03c1\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ac\u03c1\u03c4\u03b9\u03bf\u03c2 \u03ac\u03c1\u03b1 \u03b4\u03b9\u03b1\u03b9\u03c1\u03ce \u03bc\u03b5 \u03c4\u03bf 2 \u03ba\u03b1\u03b9 \u03ad\u03c7\u03c9 \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 17.\u0395\u03af\u03bd\u03b1\u03b9 \u03c0\u03b5\u03c1\u03b9\u03c4\u03c4\u03cc\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03af\u03b6\u03c9 \u03c4\u03c1\u03b9\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c4\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u03b7\u03bd \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1,3&#215;17+1=52. \u03a3\u03c5\u03bd\u03b5\u03c7\u03af\u03b6\u03c9 \u03c4\u03b7\u03bd \u03b9\u03b4\u03af\u03b1 \u03b4\u03b9\u03b1\u03b4\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af \u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03bf\u03c5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 26,13,40,20,10,5,16,8,4,2,1.<br \/>\n\u0391\u03c0\u03cc \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03b1\u03c5\u03c4\u03cc \u03ba\u03b1\u03b9 \u03bc\u03b5\u03c4\u03ac \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b5\u03c0\u03b1\u03bd\u03b1\u03bb\u03b1\u03bc\u03b2\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03b7 \u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03af\u03b1 4,2,1,4,2,1,4,2,1\u2026..<\/p>\n<p>\u038c\u03c4\u03b1\u03bd \u03c6\u03c4\u03ac\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c3\u03c4\u03bf 1 \u03b4\u03b9\u03b1\u03b4\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u03c4\u03b5\u03c1\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c4\u03b1\u03b9.<br \/>\n\u0394\u03b5\u03af\u03c4\u03b5 \u03c4\u03b7\u03bd \u03b1\u03bb\u03c5\u03c3\u03af\u03b4\u03b1 \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd:<\/p>\n<p>11 &#8211;&gt;431&#8211;&gt;7&#8211;&gt;52&#8211;&gt;26&#8211;&gt;13&#8211;&gt;40 &#8211;&gt;20 &#8211;&gt;10&#8211;&gt;5&#8211;&gt;16&#8211;&gt;8&#8211;&gt;4&#8211;&gt;2&#8211;&gt;1<\/p>\n<p>\u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03ad\u03b8\u03b7\u03ba\u03b5 \u03c4\u03bf 1937 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc Lothar Collatz, \u03bf \u03cc\u03c0\u03bf\u03b9\u03bf\u03c2 \u03b1\u03bd\u03b1\u03c1\u03c9\u03c4\u03ae\u03b8\u03b7\u03ba\u03b5:<br \/>\n\u00ab\u0398\u03b1 \u03ba\u03b1\u03c4\u03b1\u03bb\u03ae\u03b3\u03bf\u03c5\u03bc\u03b5 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c3\u03c4\u03bf 1, \u03b1\u03c0\u03cc \u03cc\u03c0\u03bf\u03b9\u03bf\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc \u03ba\u03b1\u03b9 \u03b1\u03bd \u03be\u03b5\u03ba\u03b9\u03bd\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5;\u00bb.<\/p>\n<p>\u038e\u03c3\u03c4\u03b5\u03c1\u03b1 \u03b1\u03c0\u03cc 80 \u03c7\u03c1\u03cc\u03bd\u03b9\u03b1 \u03b1\u03ba\u03cc\u03bc\u03b1 \u03b4\u03b5\u03bd \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b1\u03c0\u03ac\u03bd\u03c4\u03b7\u03c3\u03b7.<br \/>\nL.Collatz (1910\u20131990)<\/p>\n<p>\u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03ad\u03c7\u03b5\u03b9 \u03c0\u03bf\u03bb\u03bb\u03ac \u03bf\u03bd\u03cc\u03bc\u03b1\u03c4\u03b1, \u03b5\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 3n+1, \u03b5\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u039f\u03cd\u03bb\u03b1\u03bc \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03a3\u03c4\u03ac\u03bd\u03b9\u03bb\u03c3\u03bb\u03b1\u03b2 \u039f\u03cd\u03bb\u03b1\u03bc (Stanislaw Ulam), \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u039a\u03b1\u03ba\u03bf\u03c5\u03c4\u03ac\u03bd\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03a3\u03b9\u03b6\u03bf\u03cd\u03bf \u039a\u03b1\u03ba\u03bf\u03c5\u03c4\u03ac\u03bd\u03b9 (Shizuo Kakutani), \u03b5\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u0398\u03bf\u03c5\u03ad\u03b9\u03c4\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u039c\u03c0\u03c1\u03ac\u03b9\u03bd \u0398\u03bf\u03c5\u03ad\u03b9\u03c4\u03c2 (Bryan Thwaites), \u03b1\u03bb\u03b3\u03cc\u03c1\u03b9\u03b8\u03bc\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03a7\u03ac\u03c3\u03b5 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03a7\u03ad\u03bb\u03bc\u03bf\u03c5\u03c4 \u03a7\u03ac\u03c3\u03b5 (Helmut Hasse), \u03ae \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd \u03a3\u03c5\u03c1\u03b1\u03ba\u03bf\u03c5\u03c3\u03ce\u03bd. \u0397 \u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03af\u03b1 \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03c0\u03bf\u03c5 \u03b5\u03bc\u03c0\u03bb\u03ad\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03c4\u03b1\u03b9 \u03c9\u03c2 \u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03af\u03b1 \u03c7\u03b1\u03bb\u03b1\u03b6\u03b9\u03bf\u03cd (\u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03bf\u03b9 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c3\u03c5\u03bd\u03ae\u03b8\u03c9\u03c2 \u03c5\u03c0\u03cc\u03ba\u03b5\u03b9\u03bd\u03c4\u03b1\u03b9 \u03c3\u03b5 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03ad\u03c2 \u03ba\u03b1\u03c4\u03b1\u03b2\u03ac\u03c3\u03b5\u03b9\u03c2 \u03ba\u03b1\u03b9 \u03b1\u03bd\u03b1\u03b2\u03ac\u03c3\u03b5\u03b9\u03c2 \u03c3\u03b1\u03bd \u03c4\u03bf\u03c5\u03c2 \u03ba\u03cc\u03ba\u03ba\u03bf\u03c5\u03c2 \u03c4\u03bf\u03c5 \u03c7\u03b1\u03bb\u03b1\u03b6\u03b9\u03bf\u03cd \u03c3\u03b5 \u03ad\u03bd\u03b1 \u03c3\u03cd\u03bd\u03bd\u03b5\u03c6\u03bf, \u03b5\u03af\u03c4\u03b5 \u03c9\u03c2 \u03b8\u03b1\u03c5\u03bc\u03b1\u03c3\u03c4\u03bf\u03af \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af.<br \/>\n\u039f\u03b9 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03af \u03b5\u03c1\u03b5\u03c5\u03bd\u03b7\u03c4\u03ad\u03c2 \u03b5\u03b9\u03ba\u03ac\u03b6\u03bf\u03c5\u03bd \u03cc\u03c4\u03b9 \u03b7 \u03b1\u03c0\u03ac\u03bd\u03c4\u03b7\u03c3\u03b7 \u03c3\u03c4\u03bf \u03b5\u03c1\u03ce\u03c4\u03b7\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b1\u03c6\u03b1\u03c4\u03b9\u03ba\u03ae. \u039c\u03b5 \u03c4\u03b7\u03bd \u03c7\u03c1\u03ae\u03c3\u03b7 \u03c5\u03c0\u03b5\u03c1-\u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03c4\u03ce\u03bd \u03ba\u03b1\u03c4\u03cc\u03c1\u03b8\u03c9\u03c3\u03b1\u03bd \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bd \u03cc\u03c4\u03b9 \u03cc\u03bb\u03bf\u03b9 \u03bf\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03bf\u03bd 18*2^60 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac \u03ba\u03b1\u03c4\u03b1\u03bb\u03ae\u03b3\u03bf\u03c5\u03bd \u03c3\u03c4\u03bf 1.<br \/>\n\u03a4\u03b7\u03bd \u03b4\u03b5\u03ba\u03b1\u03b5\u03c4\u03af\u03b1 \u03c4\u03bf\u03c5 60,\u03bf \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03a3\u03b9\u03b6\u03bf\u03cd\u03bf \u039a\u03b1\u03ba\u03ba\u03bf\u03c5\u03c4\u03ac\u03bd\u03b9, \u03b4\u03b9\u03ad\u03b4\u03c9\u03c3\u03b5 \u03c4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c3\u03c4\u03bf \u03c0\u03b1\u03bd\u03b5\u03c0\u03b9\u03c3\u03c4\u03ae\u03bc\u03b9\u03bf \u03c4\u03bf\u03c5 \u0393\u03b5\u03b9\u03bb. \u0395 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd, \u03b3\u03b9\u03b1 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03bf \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1\u03bd \u03bc\u03ae\u03bd\u03b1 \u03cc\u03bb\u03bf\u03b9 \u03bf\u03b9 \u03b8\u03b5\u03c1\u03ac\u03c0\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c4\u03c9\u03bd \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03c3\u03c4\u03bf \u0393\u03ad\u03b9\u03bb ,\u03ba\u03b1\u03b8\u03b7\u03b3\u03b7\u03c4\u03ad\u03c2, 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\u03b3\u03b9\u03b1 500 \u03b4\u03bf\u03bb\u03ac\u03c1\u03b9\u03b1, \u03c7\u03c9\u03c1\u03af\u03c2 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03b9\u03c9\u03b8\u03b5\u03af \u03ba\u03ac\u03c0\u03bf\u03b9\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>\u039c\u03b1\u03b8\u03b7-\u03bc\u03b1\u03b3\u03b9\u03ba\u03ac<\/strong><\/em><br \/>\n<em><strong>\u0391\u03b8.\u0394\u03a1\u039f\u03a5\u0393\u0391\u03a3<\/strong><\/em><br \/>\n<em>\u0395\u03b9\u03ba\u03cc\u03bd\u03b1: https:\/\/www.youtube.com\/watch?v=jFztIvYxnwM<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0388\u03bd\u03b1 \u03c3\u03ba\u03bb\u03b7\u03c1\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03ba\u03b1\u03b9 \u03b7 \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03b5\u03bd\u03cc\u03c2! &#8220;\u0397 \u03b1\u03c0\u03cc\u03bb\u03c5\u03c4\u03b7 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bf\u03b4\u03b7\u03b3\u03b5\u03af \u03c3\u03c4\u03b7\u03bd \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03b5\u03bd\u03cc\u03c2.&#8221; \u039c\u03bf\u03bd\u03c4\u03b5\u03c3\u03ba\u03b9\u03ad \u03a3\u03c4\u03b9\u03c2 6 \u0399\u03bf\u03c5\u03bb\u03af\u03bf\u03c5 1910, \u03b3\u03b5\u03bd\u03bd\u03ae\u03b8\u03b7\u03ba\u03b5 \u03bf \u0393\u03b5\u03c1\u03bc\u03b1\u03bd\u03cc\u03c2 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 Lothar Collatz. To 1937 \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b5 \u03bc\u03b9\u03b1 \u03b5\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u03c0\u03bf\u03c5 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9 \u03ac\u03bb\u03c5\u03c4\u03b7. \u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd \u03a3\u03c5\u03c1\u03b1\u03ba\u03bf\u03c5\u03c3\u03ce\u03bd. \u0388\u03bd\u03b1 \u03ba\u03b1\u03c4\u03ac \u03b2\u03ac\u03c3\u03b7 \u03b1\u03c0\u03bb\u03cc \u03c3\u03c4\u03b7\u03bd \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b7&#8230;<\/p>\n","protected":false},"author":1,"featured_media":27941,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1,66],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=900%2C609&ssl=1","rttpg_featured_image_url":{"full":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=900%2C609&ssl=1",900,609,false],"landscape":["https:\/\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg",900,609,false],"portraits":["https:\/\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg",900,609,false],"thumbnail":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=150%2C150&ssl=1",150,150,true],"medium":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=300%2C203&ssl=1",300,203,true],"large":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=900%2C609&ssl=1",900,609,true],"1536x1536":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=900%2C609&ssl=1",900,609,true],"2048x2048":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=900%2C609&ssl=1",900,609,true],"portfolio-square":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=570%2C570&ssl=1",570,570,true],"portfolio-portrait":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=600%2C609&ssl=1",600,609,true],"portfolio-landscape":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=800%2C600&ssl=1",800,600,true],"menu-featured-post":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=345%2C198&ssl=1",345,198,true],"qode-carousel_slider":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=400%2C260&ssl=1",400,260,true],"portfolio_slider":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=500%2C380&ssl=1",500,380,true],"portfolio_masonry_regular":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=500%2C500&ssl=1",500,500,true],"portfolio_masonry_wide":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=900%2C500&ssl=1",900,500,true],"portfolio_masonry_tall":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=500%2C609&ssl=1",500,609,true],"portfolio_masonry_large":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=900%2C609&ssl=1",900,609,true],"portfolio_masonry_with_space":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=700%2C474&ssl=1",700,474,true],"latest_post_boxes":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=539%2C303&ssl=1",539,303,true],"woocommerce_thumbnail":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=300%2C300&ssl=1",300,300,true],"woocommerce_single":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?fit=600%2C406&ssl=1",600,406,true],"woocommerce_gallery_thumbnail":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2018\/08\/post-1448.jpg?resize=100%2C100&ssl=1",100,100,true]},"rttpg_author":{"display_name":"admin","author_link":"https:\/\/www.lecturesbureau.gr\/1\/author\/admin\/"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/www.lecturesbureau.gr\/1\/category\/uncategorized\/?lang=en\" rel=\"category tag\">Uncategorized<\/a> <a href=\"https:\/\/www.lecturesbureau.gr\/1\/category\/science\/\" rel=\"category tag\">\u0395\u03a0\u0399\u03a3\u03a4\u0397\u039c\u0397<\/a>","rttpg_excerpt":"\u0388\u03bd\u03b1 \u03c3\u03ba\u03bb\u03b7\u03c1\u03cc \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03ba\u03b1\u03b9 \u03b7 \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03b5\u03bd\u03cc\u03c2! &#8220;\u0397 \u03b1\u03c0\u03cc\u03bb\u03c5\u03c4\u03b7 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bf\u03b4\u03b7\u03b3\u03b5\u03af \u03c3\u03c4\u03b7\u03bd \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03b5\u03bd\u03cc\u03c2.&#8221; \u039c\u03bf\u03bd\u03c4\u03b5\u03c3\u03ba\u03b9\u03ad \u03a3\u03c4\u03b9\u03c2 6 \u0399\u03bf\u03c5\u03bb\u03af\u03bf\u03c5 1910, \u03b3\u03b5\u03bd\u03bd\u03ae\u03b8\u03b7\u03ba\u03b5 \u03bf \u0393\u03b5\u03c1\u03bc\u03b1\u03bd\u03cc\u03c2 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 Lothar Collatz. To 1937 \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b5 \u03bc\u03b9\u03b1 \u03b5\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u03c0\u03bf\u03c5 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9 \u03ac\u03bb\u03c5\u03c4\u03b7. \u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd \u03a3\u03c5\u03c1\u03b1\u03ba\u03bf\u03c5\u03c3\u03ce\u03bd. \u0388\u03bd\u03b1 \u03ba\u03b1\u03c4\u03ac \u03b2\u03ac\u03c3\u03b7 \u03b1\u03c0\u03bb\u03cc \u03c3\u03c4\u03b7\u03bd \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b7...","_links":{"self":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts\/27949"}],"collection":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/comments?post=27949"}],"version-history":[{"count":1,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts\/27949\/revisions"}],"predecessor-version":[{"id":27950,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts\/27949\/revisions\/27950"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/media\/27941"}],"wp:attachment":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/media?parent=27949"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/categories?post=27949"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/tags?post=27949"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}