Zahlentheorie
Angelegt Donnerstag 02 März 2023
Welche ganzen Zahlen kann man dadurch erzeugen, dass man die vier Grundrechenarten +,-,* und / auf die neun Ziffern 1..9 anwendet? Jede der Ziffern darf nur einmal in dieser Reihenfolge vorkommen: 1 2 3 4 5 6 7 8 9
Beispiel: 1+2-3*4/5+6-7*8+9 oder 1-2/3*4+5/6-7*8-9
Ein Programm das diese Ergebnisse berechnet:
#!/usr/bin/python3
def IstGanz(x):
return abs(x - int(x)) == 0
op = ['+','-','*','/']
numbers = []
print(op[0])
print(op[1])
print(op[2])
print(op[3])
for i in range(4):
result = eval(ausdruck)
if IstGanz(result):
numbers.sort()
print(numbers)
Resultat:
+
-
*
/
[-362879, -181441, -181438, -181437, -60485, -60484, -60481, -60480, -60478, -60475, -60474, -60474, -60473, -40328, -40319, -40310, -22679, -20170, -20167, -20166, -20152, -20149, -20148, -15143, -15133, -15130, -15129, -15129, -15128, -15125, -15124, -15122, -15121, -15120, -15119, -15118, -15118, -15117, -15117, -15116, -15114, -15114, -15111, -15110, -15110, -15109, -15109, -15106, -15105, -15096, -15095, -11341, -11338, -11337, -10079, -6734, -6733, -6730, -6729, -6727, -6724, -6723, -6723, -6722, -6716, -6715, -6712, -6711, -6709, -6706, -6705, -6705, -6704, -5669, -5111, -5056, -5041, -5040, -5038, -5038, -5037, -5022, -4967, -4488, -4479, -4470, -3143, -3085, -3082, -3081, -3052, -3049, -3048, -3045, -3044, -3042, -3042, -3042, -3039, -3039, -3038, -3038, -3038, -3038, -3037, -3037, -3034, -3033, -3032, -3031, -3030, -3029, -3029, -3028, -3028, -3028, -3027, -3027, -3027, -3026, -3026, -3025, -3024, -3023, -3023, -3023, -3021, -3020, -3020, -3019, -3019, -3019, -3018, -3018, 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39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 83, 83, 83, 83, 83, 83, 83, 83, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 97, 97, 97, 97, 97, 97, 97, 97, 97, 98, 98, 98, 98, 98, 98, 98, 98, 98, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 102, 102, 102, 102, 102, 102, 102, 102, 102, 103, 103, 103, 103, 103, 103, 103, 104, 104, 104, 104, 104, 104, 104, 104, 104, 104, 104, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 106, 106, 106, 106, 106, 106, 106, 106, 107, 107, 107, 107, 107, 107, 107, 108, 108, 108, 108, 108, 109, 109, 109, 109, 109, 109, 109, 109, 109, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 111, 111, 111, 111, 111, 111, 111, 111, 111, 111, 112, 112, 112, 112, 112, 112, 112, 112, 113, 113, 113, 113, 114, 114, 114, 114, 114, 115, 115, 115, 115, 115, 115, 115, 115, 115, 115, 115, 115, 116, 116, 116, 116, 116, 116, 117, 117, 117, 117, 118, 118, 118, 118, 118, 118, 118, 118, 118, 119, 119, 119, 119, 119, 119, 119, 119, 119, 119, 119, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 122, 122, 122, 122, 122, 122, 122, 123, 123, 123, 123, 123, 123, 123, 124, 124, 124, 124, 124, 124, 125, 125, 125, 125, 125, 125, 125, 126, 126, 127, 127, 128, 128, 128, 128, 128, 129, 129, 129, 129, 129, 129, 130, 130, 130, 130, 130, 130, 130, 130, 131, 132, 132, 132, 132, 132, 133, 133, 133, 133, 133, 133, 133, 133, 133, 133, 133, 134, 134, 134, 134, 134, 134, 134, 134, 134, 134, 135, 135, 135, 135, 136, 136, 136, 136, 136, 136, 137, 137, 137, 138, 138, 139, 139, 139, 139, 139, 139, 139, 140, 140, 140, 140, 140, 140, 141, 141, 141, 141, 141, 141, 141, 142, 142, 142, 143, 143, 143, 143, 144, 144, 144, 144, 144, 144, 144, 145, 145, 145, 145, 146, 146, 146, 146, 146, 147, 147, 147, 148, 148, 148, 149, 149, 149, 150, 150, 150, 150, 151, 151, 151, 152, 153, 156, 157, 157, 157, 158, 159, 159, 160, 161, 161, 162, 162, 162, 162, 163, 163, 163, 164, 166, 167, 169, 170, 170, 172, 172, 173, 173, 173, 173, 174, 174, 174, 175, 176, 177, 179, 179, 179, 180, 180, 180, 180, 180, 180, 181, 181, 183, 183, 184, 184, 184, 184, 185, 185, 185, 185, 186, 186, 187, 187, 188, 188, 188, 189, 190, 190, 190, 191, 191, 191, 191, 191, 191, 191, 191, 192, 192, 192, 192, 192, 193, 193, 194, 194, 194, 195, 195, 195, 196, 196, 196, 197, 198, 198, 199, 199, 199, 199, 200, 200, 201, 201, 201, 202, 202, 202, 203, 203, 203, 204, 204, 204, 204, 204, 204, 205, 205, 205, 205, 206, 206, 206, 207, 207, 207, 208, 208, 208, 209, 209, 210, 210, 211, 211, 211, 212, 212, 212, 212, 212, 213, 213, 213, 213, 214, 214, 214, 215, 215, 215, 216, 217, 217, 217, 217, 217, 218, 218, 218, 218, 219, 219, 219, 220, 220, 220, 221, 221, 222, 222, 222, 223, 223, 224, 225, 225, 226, 226, 226, 227, 227, 228, 229, 229, 230, 230, 230, 231, 233, 233, 233, 234, 234, 235, 235, 236, 236, 236, 236, 237, 237, 238, 238, 239, 240, 241, 242, 243, 244, 245, 245, 251, 252, 259, 259, 266, 269, 269, 270, 271, 271, 272, 272, 273, 273, 274, 277, 278, 280, 280, 280, 281, 281, 282, 283, 283, 284, 284, 284, 284, 285, 285, 286, 287, 288, 288, 288, 289, 289, 290, 291, 292, 292, 293, 293, 294, 294, 296, 297, 297, 297, 298, 298, 298, 299, 299, 302, 303, 305, 306, 306, 307, 307, 308, 309, 309, 309, 309, 312, 312, 312, 313, 313, 313, 313, 314, 314, 314, 314, 315, 315, 316, 316, 317, 317, 317, 317, 318, 318, 319, 320, 320, 321, 321, 322, 322, 323, 323, 323, 323, 324, 324, 324, 324, 325, 325, 325, 326, 326, 327, 327, 327, 327, 327, 328, 328, 328, 330, 330, 330, 331, 331, 331, 331, 331, 331, 332, 332, 332, 332, 332, 332, 333, 333, 333, 333, 334, 334, 335, 335, 335, 335, 336, 336, 336, 337, 337, 338, 338, 338, 338, 339, 339, 340, 340, 341, 341, 341, 341, 342, 342, 342, 342, 342, 342, 343, 343, 343, 343, 343, 344, 345, 346, 346, 346, 346, 346, 346, 347, 347, 347, 348, 349, 349, 349, 349, 350, 350, 350, 351, 351, 351, 351, 352, 352, 352, 352, 353, 353, 353, 353, 353, 353, 354, 354, 354, 354, 355, 355, 356, 356, 356, 356, 357, 357, 357, 359, 360, 360, 360, 361, 361, 361, 364, 364, 364, 365, 365, 365, 365, 367, 367, 368, 369, 369, 370, 370, 371, 371, 371, 372, 372, 373, 374, 375, 375, 379, 379, 380, 383, 383, 384, 386, 386, 386, 387, 387, 388, 389, 389, 390, 390, 390, 391, 391, 404, 406, 407, 408, 409, 410, 424, 424, 425, 427, 427, 428, 428, 437, 438, 440, 441, 441, 442, 447, 448, 449, 451, 452, 453, 460, 461, 464, 465, 465, 465, 466, 466, 469, 470, 470, 472, 473, 473, 474, 474, 475, 476, 476, 477, 477, 477, 478, 478, 480, 480, 480, 480, 480, 482, 482, 483, 483, 483, 484, 484, 484, 484, 484, 484, 485, 485, 485, 485, 485, 485, 486, 488, 488, 488, 489, 489, 489, 489, 490, 490, 491, 492, 492, 492, 492, 493, 493, 493, 494, 494, 494, 495, 495, 495, 495, 495, 496, 496, 496, 496, 496, 496, 496, 497, 497, 497, 497, 498, 498, 499, 499, 499, 499, 500, 500, 501, 501, 501, 501, 502, 502, 502, 503, 503, 503, 503, 504, 504, 504, 504, 504, 505, 505, 505, 505, 505, 506, 506, 506, 506, 506, 506, 507, 507, 507, 507, 507, 508, 508, 508, 508, 509, 509, 509, 509, 509, 509, 509, 510, 510, 511, 511, 511, 511, 512, 513, 513, 513, 513, 513, 514, 514, 514, 514, 514, 515, 515, 515, 516, 516, 516, 516, 517, 517, 517, 517, 517, 517, 518, 518, 518, 518, 518, 519, 519, 520, 520, 521, 521, 521, 523, 524, 524, 524, 524, 524, 524, 525, 525, 525, 525, 525, 525, 525, 526, 526, 526, 526, 527, 527, 528, 528, 529, 529, 529, 529, 530, 530, 530, 530, 532, 532, 533, 534, 535, 535, 536, 536, 536, 536, 537, 537, 537, 538, 539, 540, 540, 540, 540, 543, 543, 544, 544, 544, 545, 545, 548, 549, 549, 550, 550, 553, 554, 557, 558, 559, 559, 560, 560, 561, 561, 562, 568, 569, 571, 572, 572, 573, 577, 578, 584, 585, 618, 619, 619, 620, 621, 621, 622, 623, 624, 626, 629, 630, 630, 630, 630, 630, 630, 631, 631, 631, 631, 632, 633, 639, 639, 640, 641, 642, 655, 655, 656, 656, 673, 674, 684, 696, 697, 710, 711, 712, 713, 714, 715, 726, 727, 728, 729, 730, 731, 744, 745, 747, 763, 764, 767, 767, 768, 768, 770, 773, 774, 774, 775, 785, 785, 786, 786, 799, 800, 818, 819, 822, 823, 825, 828, 829, 829, 830, 834, 835, 836, 837, 838, 839, 840, 841, 841, 843, 844, 845, 845, 846, 846, 847, 847, 848, 852, 853, 856, 857, 859, 862, 863, 863, 863, 864, 866, 867, 907, 908, 911, 912, 914, 917, 918, 918, 919, 940, 941, 944, 945, 947, 950, 951, 951, 952, 1111, 1111, 1112, 1120, 1129, 1129, 1130, 1188, 1224, 1225, 1243, 1250, 1253, 1254, 1259, 1261, 1268, 1271, 1272, 1277, 1332, 1648, 1658, 1661, 1662, 1662, 1663, 1666, 1666, 1667, 1669, 1670, 1671, 1672, 1673, 1673, 1674, 1674, 1675, 1676, 1677, 1677, 1679, 1680, 1680, 1680, 1681, 1681, 1681, 1682, 1682, 1684, 1685, 1685, 1686, 1687, 1688, 1689, 1690, 1691, 1691, 1692, 1692, 1693, 1695, 1695, 1695, 1696, 1698, 1699, 1699, 1700, 1700, 1703, 1704, 1713, 1714, 2239, 2242, 2243, 2447, 2450, 2451, 2502, 2505, 2506, 2511, 2512, 2518, 2520, 2520, 2520, 2521, 2521, 2522, 2523, 2524, 2529, 2530, 2536, 2539, 2540, 2591, 2594, 2595, 2834, 2837, 2838, 2905, 2963, 2966, 2967, 2996, 2999, 3000, 3003, 3004, 3006, 3006, 3006, 3009, 3009, 3010, 3010, 3010, 3010, 3011, 3011, 3014, 3015, 3016, 3017, 3018, 3019, 3019, 3020, 3020, 3020, 3021, 3021, 3021, 3022, 3022, 3023, 3024, 3025, 3025, 3025, 3027, 3028, 3028, 3029, 3029, 3029, 3030, 3030, 3030, 3031, 3031, 3032, 3032, 3033, 3033, 3034, 3035, 3035, 3038, 3039, 3039, 3039, 3040, 3040, 3040, 3043, 3043, 3043, 3044, 3044, 3044, 3046, 3049, 3050, 3050, 3051, 3053, 3054, 3054, 3083, 3086, 3087, 3144, 3145, 4471, 4472, 4480, 4481, 4489, 4490, 4968, 4969, 5023, 5024, 5039, 5039, 5040, 5041, 5042, 5042, 5043, 5057, 5058, 5112, 5113, 5670, 5670, 5671, 6706, 6707, 6710, 6711, 6713, 6716, 6717, 6717, 6718, 6724, 6725, 6728, 6729, 6731, 6734, 6735, 6735, 6736, 10071, 10080, 10080, 10081, 10089, 11339, 11342, 11343, 15097, 15107, 15110, 15111, 15111, 15112, 15115, 15116, 15118, 15119, 15120, 15121, 15122, 15122, 15123, 15123, 15124, 15126, 15126, 15129, 15130, 15130, 15131, 15131, 15134, 15135, 15144, 15145, 20150, 20153, 20154, 20168, 20171, 20172, 22680, 22681, 40311, 40312, 40320, 40321, 40329, 40330, 60475, 60476, 60479, 60480, 60482, 60485, 60486, 60486, 60487, 90720, 181439, 181442, 181443, 362880, 362881]
Sehr spannend ist das Ergebnis 362881. Diese Zahl ist um 1 größer als 9! (Fakultät). Ich dachte erst, dass 1*2*3*4*5*6*7*8*9 = 362880 die größte Zahl sein müsste, aber nach kurzer Überlegung sah ich, dass 1+2*3*4*5*6*7*8*9 noch um 1 größer als 9! ist :-)
Eine Variante des Programms könnte noch die korrespondierenden Ausdrücke zu den Resultaten mit ausgeben. Dann könnte man das Programm befragen: Welche Ausdrücke ergeben 0 oder 111?
#!/usr/bin/python3
def istganz(x):
zahl = input("Zu welcher Zahl wollen Sie den Ausdruck sehen? ")
op = ['+','-','*','/']
numbers = []
flag = 1
for i in range(4):
result = eval(ausdruck)
if istganz(result) and int(zahl)==int(result):
flag = 0
if flag == 1:
Jedes Programm ist ein Löffel!
Es beginnt mit einer Initialisierung, welche nur einmal durchlaufen wird. Dann schließt sich der Loop an, der entweder niemals endet oder irgendwann abbricht. Das man das Abbrechen nicht im Allgemeinen für jedes beliebige Programm im voraus berechnen kann, nennt man das Halteproblem. Aber Achtung! Für jedes spezielle Programm kann man sich die Mühe machen und untersuchen, ob es halten wird. Man kann nur nicht die eine Software schreiben, die das für jedes beliebige Programm entscheidet.
Lemma: Wenn man "Programm" mit "Problem" gleichsetzt, dann folgt daraus, dass es unlösbare Probleme gibt.
Lemma: Die meisten Probleme lösen sich von selbst, wenn man sie nicht dabei stört.