{"id":249,"date":"2025-11-17T18:13:42","date_gmt":"2025-11-17T18:13:42","guid":{"rendered":"https:\/\/maths4u.gr\/?page_id=249"},"modified":"2025-12-15T17:16:48","modified_gmt":"2025-12-15T15:16:48","slug":"perfect_numbers","status":"publish","type":"page","link":"https:\/\/maths4u.gr\/index.php\/perfect_numbers\/","title":{"rendered":"\u03a4\u03ad\u03bb\u03b5\u03b9\u03bf\u03b9 \u0391\u03c1\u03b9\u03b8\u03bc\u03bf\u03af"},"content":{"rendered":"<p>\u0388\u03bd\u03b1\u03c2 \u03c6\u03c5\u03c3\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03bb\u03ad\u03b3\u03b5\u03c4\u03b1\u03b9 <strong>\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2<\/strong> \u03b1\u03bd \u03b9\u03c3\u03bf\u03cd\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03c4\u03bf\u03c5 \u03b5\u03ba\u03c4\u03cc\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03af\u03b4\u03b9\u03bf, \u03c0.\u03c7. 6=1+2+3.<br \/>\n\u039f \u0395\u03c5\u03ba\u03bb\u03b5\u03af\u03b4\u03b7\u03c2 \u03c3\u03c4\u03b1 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03c0\u03b5\u03c1\u03b9\u03b3\u03c1\u03ac\u03c6\u03b5\u03b9 \u03c4\u03bf\u03c5\u03c2 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c5\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03ba\u03b1\u03b9 \u03b4\u03af\u03bd\u03b5\u03b9 \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc \u03c4\u03cd\u03c0\u03bf. \u039c\u03b1\u03c2 \u03bb\u03ad\u03b5\u03b9 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03cc\u03c4\u03b9 \u03b1\u03bd \u03bf \\(p\\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1\u03c5\u03c4\u03cc\u03c7\u03c1\u03bf\u03bd\u03b1 \u03bf \\( 2^p &#8211; 1 \\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2, \u03c4\u03cc\u03c4\u03b5 \u03bf \\( N = 2^{p-1}(2^p &#8211; 1) \\) \u03b5\u03af\u03bd\u03b1\u03b9 <strong>\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2<\/strong>.<\/p>\n<p><strong>\u03a0\u03b1\u03c1\u03b1\u03b4\u03b5\u03af\u03b3\u03bc\u03b1\u03c4\u03b1<\/strong><\/p>\n<table>\n<thead>\n<tr>\n<td><strong>(p)<\/strong><\/td>\n<td><strong>\\(2^p &#8211; 1\\)<\/strong><\/td>\n<td><strong>\u03a4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \\(N = 2^{p-1}(2^p &#8211; 1)\\)<\/strong><\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>3 (\u03c0\u03c1\u03ce\u03c4\u03bf\u03c2)<\/td>\n<td>\\(2^{1} \\times 3 = 6\\)<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7 (\u03c0\u03c1\u03ce\u03c4\u03bf\u03c2)<\/td>\n<td>\\(2^{2} \\times 7 = 28\\)<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>31 (\u03c0\u03c1\u03ce\u03c4\u03bf\u03c2)<\/td>\n<td>\\(2^{4} \\times 31 = 496\\)<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>127 (\u03c0\u03c1\u03ce\u03c4\u03bf\u03c2)<\/td>\n<td>\\(2^{6} \\times 127 = 8128\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0391\u03c2 \u03b4\u03bf\u03cd\u03bc\u03b5 \u03c4\u03ce\u03c1\u03b1 \u03b3\u03b9\u03b1\u03c4\u03af \u03bf \u03c4\u03cd\u03c0\u03bf\u03c2 \u03c4\u03bf\u03c5 \u0395\u03c5\u03ba\u03bb\u03b5\u03af\u03b4\u03b7<br \/>\n\\[<br \/>\nN = 2^{p-1}(2^p &#8211; 1)<br \/>\n\\]<br \/>\n\u03b4\u03af\u03bd\u03b5\u03b9 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b9 <strong>\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc<\/strong>, \u03cc\u03c4\u03b1\u03bd \\(2^p &#8211; 1\\) \u03b5\u03af\u03bd\u03b1\u03b9 <strong>\u03c0\u03c1\u03ce\u03c4\u03bf\u03c2<\/strong>.<\/p>\n<p><strong>\ud83d\udd39<\/strong><strong> \u03a5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7:<\/strong><\/p>\n<p>\u0388\u03c3\u03c4\u03c9 \u03cc\u03c4\u03b9 \\(p\\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2 \u03ba\u03b1\u03b9 \\(2^p &#8211; 1\\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2 (\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae <strong>\u03c0\u03c1\u03ce\u03c4\u03bf\u03c2 \u03c4\u03bf\u03c5 \u039c\u03b5\u03c1\u03c3\u03ad\u03bd<\/strong>).<\/p>\n<p>\u0398\u03ad\u03bb\u03bf\u03c5\u03bc\u03b5 \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9<br \/>\n\\[<br \/>\nN = 2^{p-1}(2^p &#8211; 1)<br \/>\n\\]<br \/>\n\u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03cc\u03c4\u03b9 \u03c4\u03bf <strong>\u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03cc\u03bb\u03c9\u03bd \u03c4\u03c9\u03bd \u03b8\u03b5\u03c4\u03b9\u03ba\u03ce\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03c4\u03bf\u03c5<\/strong> (\u03b5\u03ba\u03c4\u03cc\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03af\u03b4\u03b9\u03bf) \u03b9\u03c3\u03bf\u03cd\u03c4\u03b1\u03b9 \u03bc\u03b5 (N).<\/p>\n<p><strong>\ud83d\udd39<\/strong><strong> \u0392\u03ae\u03bc\u03b1 1: \u039f\u03b9 \u03b4\u03b9\u03b1\u03b9\u03c1\u03ad\u03c4\u03b5\u03c2 \u03c4\u03bf\u03c5 \\(N\\)<\/strong><\/p>\n<p>\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \\(2^{p-1}\\) \u03ba\u03b1\u03b9 \\(2^p &#8211; 1\\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03b1\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2 (\u03b4\u03b5\u03bd \u03ad\u03c7\u03bf\u03c5\u03bd \u03ba\u03bf\u03b9\u03bd\u03bf\u03cd\u03c2 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b5\u03c2),<br \/>\n\u03bf\u03b9 \u03b4\u03b9\u03b1\u03b9\u03c1\u03ad\u03c4\u03b5\u03c2 \u03c4\u03bf\u03c5 \\(N\\) \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03bf\u03c5\u03bd \u03b1\u03bd \u03c0\u03ac\u03c1\u03bf\u03c5\u03bc\u03b5:<\/p>\n<ul>\n<li>\u03ad\u03bd\u03b1\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03ad\u03c4\u03b7 \u03c4\u03bf\u03c5 \\(2^{p-1}\\), \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03ba\u03ac\u03c0\u03bf\u03b9\u03bf \u03b1\u03c0\u03cc \u03c4\u03bf\u03c5\u03c2 \\(1, 2, 2^2, \\ldots, 2^{p-1}\\),<\/li>\n<li>\u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03b1\u03b9\u03c1\u03b5\u03c4\u03b9\u03ba\u03ac, \u03bd\u03b1 \u03c4\u03bf\u03bd \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03c3\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 \\(2^p &#8211; 1\\).<\/li>\n<\/ul>\n<p>\u0386\u03c1\u03b1 \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03b5\u03af\u03bd\u03b1\u03b9:<br \/>\n\\[<br \/>\n1, 2, 2^2, \\ldots, 2^{p-1}, \\quad (2^p &#8211; 1), 2(2^p &#8211; 1), 2^2(2^p &#8211; 1), \\ldots, 2^{p-1}(2^p &#8211; 1)<br \/>\n\\]<\/p>\n<p><strong>\ud83d\udd39<\/strong><strong> \u0392\u03ae\u03bc\u03b1 2: \u0386\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd<\/strong><\/p>\n<p>\u03a4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03b5\u03bd\u03cc\u03c2 \u03b3\u03b9\u03bd\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 \\(a \\cdot b\\) (\u03cc\u03c4\u03b1\u03bd \u03b4\u03b5\u03bd \u03ad\u03c7\u03bf\u03c5\u03bd \u03ba\u03bf\u03b9\u03bd\u03bf\u03cd\u03c2 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b5\u03c2) \u03b5\u03af\u03bd\u03b1\u03b9<br \/>\n\\[<br \/>\n\\sigma(a b) = \\sigma(a)\\sigma(b)<br \/>\n\\]<br \/>\n\u03cc\u03c0\u03bf\u03c5 \\(\\sigma(x)\\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03c4\u03bf\u03c5 \\(x\\).<\/p>\n<p>\u0395\u03b4\u03ce \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd:<br \/>\n\\[<br \/>\n\\sigma(N) = \\sigma(2^{p-1}) \\cdot \\sigma(2^p &#8211; 1)<br \/>\n\\]<\/p>\n<p><strong>\ud83d\udd39<\/strong><strong> \u0392\u03ae\u03bc\u03b1 3: \u03a5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1 \u03c7\u03c9\u03c1\u03b9\u03c3\u03c4\u03ac<\/strong><\/p>\n<p>1\ufe0f\u20e3 \u0393\u03b9\u03b1 \\(2^{p-1}\\):<br \/>\n\\[<br \/>\n\\sigma(2^{p-1}) = 1 + 2 + 2^2 + \\cdots + 2^{p-1} = 2^p &#8211; 1<br \/>\n\\]<\/p>\n<p>2\ufe0f\u20e3 \u0393\u03b9\u03b1 \\(2^p &#8211; 1\\) (\u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2, \u03ac\u03c1\u03b1 \u03bf\u03b9 \u03b4\u03b9\u03b1\u03b9\u03c1\u03ad\u03c4\u03b5\u03c2 \u03c4\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 1 \u03ba\u03b1\u03b9 \u03bf \u03af\u03b4\u03b9\u03bf\u03c2):<br \/>\n\\[<br \/>\n\\sigma(2^p &#8211; 1) = 1 + (2^p &#8211; 1) = 2^p<br \/>\n\\]<\/p>\n<p><strong>\ud83d\udd39<\/strong><strong> \u0392\u03ae\u03bc\u03b1 4: \u03a3\u03c5\u03bd\u03b4\u03c5\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5<\/strong><\/p>\n<p>\\[<br \/>\n\\sigma(N) = (2^p &#8211; 1) \\cdot 2^p = 2^p(2^p &#8211; 1)<br \/>\n\\]<\/p>\n<p>\u0391\u03bb\u03bb\u03ac \u03bf \u03af\u03b4\u03b9\u03bf\u03c2 \u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9:<br \/>\n\\[<br \/>\nN = 2^{p-1}(2^p &#8211; 1)<br \/>\n\\]<\/p>\n<p><strong>\ud83d\udd39<\/strong><strong> \u0392\u03ae\u03bc\u03b1 5: \u03a3\u03c5\u03bc\u03c0\u03ad\u03c1\u03b1\u03c3\u03bc\u03b1<\/strong><\/p>\n<p>\u03a4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 <strong>\u03cc\u03bb\u03c9\u03bd \u03c4\u03c9\u03bd<\/strong> \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \\(2 \\times N\\).<br \/>\n\u0386\u03c1\u03b1 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 <strong>\u03c4\u03c9\u03bd \u03ba\u03b1\u03c4\u03ac\u03bb\u03bb\u03b7\u03bb\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd<\/strong> (\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03c7\u03c9\u03c1\u03af\u03c2 \u03c4\u03bf\u03bd \u03af\u03b4\u03b9\u03bf \u03c4\u03bf\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc) \u03b5\u03af\u03bd\u03b1\u03b9:<\/p>\n<p>\\[<br \/>\n\\sigma(N) &#8211; N = 2N &#8211; N = N<br \/>\n\\]<\/p>\n<p>\u0386\u03c1\u03b1 \u03bf \\(N\\) \u03b5\u03af\u03bd\u03b1\u03b9 <strong>\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0388\u03bd\u03b1\u03c2 \u03c6\u03c5\u03c3\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03bb\u03ad\u03b3\u03b5\u03c4\u03b1\u03b9 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2 \u03b1\u03bd \u03b9\u03c3\u03bf\u03cd\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03ce\u03bd \u03c4\u03bf\u03c5 \u03b5\u03ba\u03c4\u03cc\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03af\u03b4\u03b9\u03bf, \u03c0.\u03c7. 6=1+2+3. \u039f \u0395\u03c5\u03ba\u03bb\u03b5\u03af\u03b4\u03b7\u03c2 \u03c3\u03c4\u03b1 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03c0\u03b5\u03c1\u03b9\u03b3\u03c1\u03ac\u03c6\u03b5\u03b9 \u03c4\u03bf\u03c5\u03c2 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c5\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03ba\u03b1\u03b9 \u03b4\u03af\u03bd\u03b5\u03b9 \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc \u03c4\u03cd\u03c0\u03bf. \u039c\u03b1\u03c2 \u03bb\u03ad\u03b5\u03b9 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03cc\u03c4\u03b9 \u03b1\u03bd \u03bf \\(p\\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1\u03c5\u03c4\u03cc\u03c7\u03c1\u03bf\u03bd\u03b1 \u03bf \\( 2^p &#8211; 1 \\) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c2, \u03c4\u03cc\u03c4\u03b5 \u03bf \\( N &hellip; <a href=\"https:\/\/maths4u.gr\/index.php\/perfect_numbers\/\" class=\"more-link\">\u0394\u03b9\u03b1\u03b2\u03ac\u03c3\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b1<span class=\"screen-reader-text\"> &#8220;\u03a4\u03ad\u03bb\u03b5\u03b9\u03bf\u03b9 \u0391\u03c1\u03b9\u03b8\u03bc\u03bf\u03af&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-249","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/pages\/249","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/comments?post=249"}],"version-history":[{"count":11,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/pages\/249\/revisions"}],"predecessor-version":[{"id":450,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/pages\/249\/revisions\/450"}],"wp:attachment":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/media?parent=249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}