Die Lösung des Luzifer-Rätsels
(Norbert Schultheis)
Rubrik: Goodies
Das oft so genannte Luzifer-Rätsel des niederländischen Mathematikers Hans Freudenthal ist ein eindrucksvolles Beispiel, wie durch einfache und klar verständliche Formulierungen nicht nur ein schwieriges mathematisches Rätsel vermittelt werden kann, sondern gleichzeitig auch eindeutige Hinweise auf dessen Lösung impliziert werden können.
Hier eine Formulierung des Rätsels:
Die berühmten Mathematiker Carl Friedrich Gauß und Leonhard Euler landen nach ihrem Tod in der Hölle. Luzifer verspricht ihnen die Freiheit, wenn sie die beiden ganzen Zahlen größer als 1 und kleiner als 100 erraten, die er sich ausgedacht hat. Er nennt Gauß das Produkt und Euler die Summe der beiden Zahlen; darauf entwickelt sich zwischen den Mathematikern folgender Dialog: Gauß: „Ich kann die zwei Zahlen nicht finden!“ Euler: „Ja, das war mir klar.“ Gauß: „Ah, jetzt kenne ich die beiden Zahlen.“ Euler: „Also, dann kenne ich sie jetzt auch.“ Die Lösung des Rätsels ergibt sich allein aus den Hinweisen im Dialog. Um dies zu verdeutlichen, betrachte ich nun die einzelnen Aussagen der beiden Mathematiker und zeige, welche Einschränkungen sich daraus ergeben, ohne dabei in mathematisches Fachjargon zu entgleiten. Die Zwischenschritte, das Endergebnis und die Gegenprobe werden dabei allesamt programmiertechnisch erzeugt. 1) Gauß: „Ich kann die zwei Zahlen nicht finden!“
Gauß wurde also ein Produkt genannt, das sich durch mehr als ein Faktorenpaar bilden lässt, sonst könnte er die Lösung direkt nennen. Die sich aus den jeweils möglichen Faktorenpaaren ergebenen Summen werden in rot dargestellt - diese brauchen wir noch später. Ihm konnten also folgende Produkte genannt worden sein:
12: [2,6] -> 8 [3,4] -> 7
16: [2,8] -> 10 [4,4] -> 8
18: [2,9] -> 11 [3,6] -> 9
20: [2,10] -> 12 [4,5] -> 9
24: [2,12] -> 14 [3,8] -> 11 [4,6] -> 10
28: [2,14] -> 16 [4,7] -> 11
30: [2,15] -> 17 [3,10] -> 13 [5,6] -> 11
32: [2,16] -> 18 [4,8] -> 12
36: [2,18] -> 20 [3,12] -> 15 [4,9] -> 13 [6,6] -> 12
40: [2,20] -> 22 [4,10] -> 14 [5,8] -> 13
42: [2,21] -> 23 [3,14] -> 17 [6,7] -> 13
44: [2,22] -> 24 [4,11] -> 15
45: [3,15] -> 18 [5,9] -> 14
48: [2,24] -> 26 [3,16] -> 19 [4,12] -> 16 [6,8] -> 14
50: [2,25] -> 27 [5,10] -> 15
52: [2,26] -> 28 [4,13] -> 17
54: [2,27] -> 29 [3,18] -> 21 [6,9] -> 15
56: [2,28] -> 30 [4,14] -> 18 [7,8] -> 15
60: [2,30] -> 32 [3,20] -> 23 [4,15] -> 19 [5,12] -> 17 [6,10] -> 16
63: [3,21] -> 24 [7,9] -> 16
64: [2,32] -> 34 [4,16] -> 20 [8,8] -> 16
66: [2,33] -> 35 [3,22] -> 25 [6,11] -> 17
68: [2,34] -> 36 [4,17] -> 21
70: [2,35] -> 37 [5,14] -> 19 [7,10] -> 17
72: [2,36] -> 38 [3,24] -> 27 [4,18] -> 22 [6,12] -> 18 [8,9] -> 17
75: [3,25] -> 28 [5,15] -> 20
76: [2,38] -> 40 [4,19] -> 23
78: [2,39] -> 41 [3,26] -> 29 [6,13] -> 19
80: [2,40] -> 42 [4,20] -> 24 [5,16] -> 21 [8,10] -> 18
81: [3,27] -> 30 [9,9] -> 18
84: [2,42] -> 44 [3,28] -> 31 [4,21] -> 25 [6,14] -> 20 [7,12] -> 19
88: [2,44] -> 46 [4,22] -> 26 [8,11] -> 19
90: [2,45] -> 47 [3,30] -> 33 [5,18] -> 23 [6,15] -> 21 [9,10] -> 19
92: [2,46] -> 48 [4,23] -> 27
96: [2,48] -> 50 [3,32] -> 35 [4,24] -> 28 [6,16] -> 22 [8,12] -> 20
98: [2,49] -> 51 [7,14] -> 21
99: [3,33] -> 36 [9,11] -> 20
100: [2,50] -> 52 [4,25] -> 29 [5,20] -> 25 [10,10] -> 20
102: [2,51] -> 53 [3,34] -> 37 [6,17] -> 23
104: [2,52] -> 54 [4,26] -> 30 [8,13] -> 21
105: [3,35] -> 38 [5,21] -> 26 [7,15] -> 22
108: [2,54] -> 56 [3,36] -> 39 [4,27] -> 31 [6,18] -> 24 [9,12] -> 21
110: [2,55] -> 57 [5,22] -> 27 [10,11] -> 21
112: [2,56] -> 58 [4,28] -> 32 [7,16] -> 23 [8,14] -> 22
114: [2,57] -> 59 [3,38] -> 41 [6,19] -> 25
116: [2,58] -> 60 [4,29] -> 33
117: [3,39] -> 42 [9,13] -> 22
120: [2,60] -> 62 [3,40] -> 43 [4,30] -> 34 [5,24] -> 29 [6,20] -> 26 [8,15] -> 23 [10,12] -> 22
124: [2,62] -> 64 [4,31] -> 35
126: [2,63] -> 65 [3,42] -> 45 [6,21] -> 27 [7,18] -> 25 [9,14] -> 23
128: [2,64] -> 66 [4,32] -> 36 [8,16] -> 24
130: [2,65] -> 67 [5,26] -> 31 [10,13] -> 23
132: [2,66] -> 68 [3,44] -> 47 [4,33] -> 37 [6,22] -> 28 [11,12] -> 23
135: [3,45] -> 48 [5,27] -> 32 [9,15] -> 24
136: [2,68] -> 70 [4,34] -> 38 [8,17] -> 25
138: [2,69] -> 71 [3,46] -> 49 [6,23] -> 29
140: [2,70] -> 72 [4,35] -> 39 [5,28] -> 33 [7,20] -> 27 [10,14] -> 24
144: [2,72] -> 74 [3,48] -> 51 [4,36] -> 40 [6,24] -> 30 [8,18] -> 26 [9,16] -> 25 [12,12] -> 24
147: [3,49] -> 52 [7,21] -> 28
148: [2,74] -> 76 [4,37] -> 41
150: [2,75] -> 77 [3,50] -> 53 [5,30] -> 35 [6,25] -> 31 [10,15] -> 25
152: [2,76] -> 78 [4,38] -> 42 [8,19] -> 27
153: [3,51] -> 54 [9,17] -> 26
154: [2,77] -> 79 [7,22] -> 29 [11,14] -> 25
156: [2,78] -> 80 [3,52] -> 55 [4,39] -> 43 [6,26] -> 32 [12,13] -> 25
160: [2,80] -> 82 [4,40] -> 44 [5,32] -> 37 [8,20] -> 28 [10,16] -> 26
162: [2,81] -> 83 [3,54] -> 57 [6,27] -> 33 [9,18] -> 27
164: [2,82] -> 84 [4,41] -> 45
165: [3,55] -> 58 [5,33] -> 38 [11,15] -> 26
168: [2,84] -> 86 [3,56] -> 59 [4,42] -> 46 [6,28] -> 34 [7,24] -> 31 [8,21] -> 29 [12,14] -> 26
170: [2,85] -> 87 [5,34] -> 39 [10,17] -> 27
171: [3,57] -> 60 [9,19] -> 28
172: [2,86] -> 88 [4,43] -> 47
174: [2,87] -> 89 [3,58] -> 61 [6,29] -> 35
175: [5,35] -> 40 [7,25] -> 32
176: [2,88] -> 90 [4,44] -> 48 [8,22] -> 30 [11,16] -> 27
180: [2,90] -> 92 [3,60] -> 63 [4,45] -> 49 [5,36] -> 41 [6,30] -> 36 [9,20] -> 29 [10,18] -> 28 [12,15] -> 27
182: [2,91] -> 93 [7,26] -> 33 [13,14] -> 27
184: [2,92] -> 94 [4,46] -> 50 [8,23] -> 31
186: [2,93] -> 95 [3,62] -> 65 [6,31] -> 37
188: [2,94] -> 96 [4,47] -> 51
189: [3,63] -> 66 [7,27] -> 34 [9,21] -> 30
190: [2,95] -> 97 [5,38] -> 43 [10,19] -> 29
192: [2,96] -> 98 [3,64] -> 67 [4,48] -> 52 [6,32] -> 38 [8,24] -> 32 [12,16] -> 28
195: [3,65] -> 68 [5,39] -> 44 [13,15] -> 28
196: [2,98] -> 100 [4,49] -> 53 [7,28] -> 35 [14,14] -> 28
198: [2,99] -> 101 [3,66] -> 69 [6,33] -> 39 [9,22] -> 31 [11,18] -> 29
200: [4,50] -> 54 [5,40] -> 45 [8,25] -> 33 [10,20] -> 30
204: [3,68] -> 71 [4,51] -> 55 [6,34] -> 40 [12,17] -> 29
207: [3,69] -> 72 [9,23] -> 32
208: [4,52] -> 56 [8,26] -> 34 [13,16] -> 29
210: [3,70] -> 73 [5,42] -> 47 [6,35] -> 41 [7,30] -> 37 [10,21] -> 31 [14,15] -> 29
216: [3,72] -> 75 [4,54] -> 58 [6,36] -> 42 [8,27] -> 35 [9,24] -> 33 [12,18] -> 30
220: [4,55] -> 59 [5,44] -> 49 [10,22] -> 32 [11,20] -> 31
222: [3,74] -> 77 [6,37] -> 43
224: [4,56] -> 60 [7,32] -> 39 [8,28] -> 36 [14,16] -> 30
225: [3,75] -> 78 [5,45] -> 50 [9,25] -> 34 [15,15] -> 30
228: [3,76] -> 79 [4,57] -> 61 [6,38] -> 44 [12,19] -> 31
230: [5,46] -> 51 [10,23] -> 33
231: [3,77] -> 80 [7,33] -> 40 [11,21] -> 32
232: [4,58] -> 62 [8,29] -> 37
234: [3,78] -> 81 [6,39] -> 45 [9,26] -> 35 [13,18] -> 31
238: [7,34] -> 41 [14,17] -> 31
240: [3,80] -> 83 [4,60] -> 64 [5,48] -> 53 [6,40] -> 46 [8,30] -> 38 [10,24] -> 34 [12,20] -> 32 [15,16] -> 31
243: [3,81] -> 84 [9,27] -> 36
245: [5,49] -> 54 [7,35] -> 42
246: [3,82] -> 85 [6,41] -> 47
248: [4,62] -> 66 [8,31] -> 39
250: [5,50] -> 55 [10,25] -> 35
252: [3,84] -> 87 [4,63] -> 67 [6,42] -> 48 [7,36] -> 43 [9,28] -> 37 [12,21] -> 33 [14,18] -> 32
255: [3,85] -> 88 [5,51] -> 56 [15,17] -> 32
256: [4,64] -> 68 [8,32] -> 40 [16,16] -> 32
258: [3,86] -> 89 [6,43] -> 49
260: [4,65] -> 69 [5,52] -> 57 [10,26] -> 36 [13,20] -> 33
261: [3,87] -> 90 [9,29] -> 38
264: [3,88] -> 91 [4,66] -> 70 [6,44] -> 50 [8,33] -> 41 [11,24] -> 35 [12,22] -> 34
266: [7,38] -> 45 [14,19] -> 33
270: [3,90] -> 93 [5,54] -> 59 [6,45] -> 51 [9,30] -> 39 [10,27] -> 37 [15,18] -> 33
272: [4,68] -> 72 [8,34] -> 42 [16,17] -> 33
273: [3,91] -> 94 [7,39] -> 46 [13,21] -> 34
275: [5,55] -> 60 [11,25] -> 36
276: [3,92] -> 95 [4,69] -> 73 [6,46] -> 52 [12,23] -> 35
279: [3,93] -> 96 [9,31] -> 40
280: [4,70] -> 74 [5,56] -> 61 [7,40] -> 47 [8,35] -> 43 [10,28] -> 38 [14,20] -> 34
282: [3,94] -> 97 [6,47] -> 53
285: [3,95] -> 98 [5,57] -> 62 [15,19] -> 34
286: [11,26] -> 37 [13,22] -> 35
288: [3,96] -> 99 [4,72] -> 76 [6,48] -> 54 [8,36] -> 44 [9,32] -> 41 [12,24] -> 36 [16,18] -> 34
290: [5,58] -> 63 [10,29] -> 39
294: [3,98] -> 101 [6,49] -> 55 [7,42] -> 49 [14,21] -> 35
296: [4,74] -> 78 [8,37] -> 45
297: [3,99] -> 102 [9,33] -> 42 [11,27] -> 38
300: [4,75] -> 79 [5,60] -> 65 [6,50] -> 56 [10,30] -> 40 [12,25] -> 37 [15,20] -> 35
304: [4,76] -> 80 [8,38] -> 46 [16,19] -> 35
306: [6,51] -> 57 [9,34] -> 43 [17,18] -> 35
308: [4,77] -> 81 [7,44] -> 51 [11,28] -> 39 [14,22] -> 36
310: [5,62] -> 67 [10,31] -> 41
312: [4,78] -> 82 [6,52] -> 58 [8,39] -> 47 [12,26] -> 38 [13,24] -> 37
315: [5,63] -> 68 [7,45] -> 52 [9,35] -> 44 [15,21] -> 36
320: [4,80] -> 84 [5,64] -> 69 [8,40] -> 48 [10,32] -> 42 [16,20] -> 36
322: [7,46] -> 53 [14,23] -> 37
324: [4,81] -> 85 [6,54] -> 60 [9,36] -> 45 [12,27] -> 39 [18,18] -> 36
325: [5,65] -> 70 [13,25] -> 38
328: [4,82] -> 86 [8,41] -> 49
330: [5,66] -> 71 [6,55] -> 61 [10,33] -> 43 [11,30] -> 41 [15,22] -> 37
336: [4,84] -> 88 [6,56] -> 62 [7,48] -> 55 [8,42] -> 50 [12,28] -> 40 [14,24] -> 38 [16,21] -> 37
340: [4,85] -> 89 [5,68] -> 73 [10,34] -> 44 [17,20] -> 37
342: [6,57] -> 63 [9,38] -> 47 [18,19] -> 37
344: [4,86] -> 90 [8,43] -> 51
345: [5,69] -> 74 [15,23] -> 38
348: [4,87] -> 91 [6,58] -> 64 [12,29] -> 41
350: [5,70] -> 75 [7,50] -> 57 [10,35] -> 45 [14,25] -> 39
351: [9,39] -> 48 [13,27] -> 40
352: [4,88] -> 92 [8,44] -> 52 [11,32] -> 43 [16,22] -> 38
357: [7,51] -> 58 [17,21] -> 38
360: [4,90] -> 94 [5,72] -> 77 [6,60] -> 66 [8,45] -> 53 [9,40] -> 49 [10,36] -> 46 [12,30] -> 42 [15,24] -> 39 [18,20] -> 38
364: [4,91] -> 95 [7,52] -> 59 [13,28] -> 41 [14,26] -> 40
368: [4,92] -> 96 [8,46] -> 54 [16,23] -> 39
370: [5,74] -> 79 [10,37] -> 47
372: [4,93] -> 97 [6,62] -> 68 [12,31] -> 43
374: [11,34] -> 45 [17,22] -> 39
375: [5,75] -> 80 [15,25] -> 40
376: [4,94] -> 98 [8,47] -> 55
378: [6,63] -> 69 [7,54] -> 61 [9,42] -> 51 [14,27] -> 41 [18,21] -> 39
380: [4,95] -> 99 [5,76] -> 81 [10,38] -> 48 [19,20] -> 39
384: [4,96] -> 100 [6,64] -> 70 [8,48] -> 56 [12,32] -> 44 [16,24] -> 40
385: [5,77] -> 82 [7,55] -> 62 [11,35] -> 46
390: [5,78] -> 83 [6,65] -> 71 [10,39] -> 49 [13,30] -> 43 [15,26] -> 41
392: [4,98] -> 102 [7,56] -> 63 [8,49] -> 57 [14,28] -> 42
396: [4,99] -> 103 [6,66] -> 72 [9,44] -> 53 [11,36] -> 47 [12,33] -> 45 [18,22] -> 40
399: [7,57] -> 64 [19,21] -> 40
400: [5,80] -> 85 [8,50] -> 58 [10,40] -> 50 [16,25] -> 41 [20,20] -> 40
405: [5,81] -> 86 [9,45] -> 54 [15,27] -> 42
406: [7,58] -> 65 [14,29] -> 43
408: [6,68] -> 74 [8,51] -> 59 [12,34] -> 46 [17,24] -> 41
410: [5,82] -> 87 [10,41] -> 51
414: [6,69] -> 75 [9,46] -> 55 [18,23] -> 41
416: [8,52] -> 60 [13,32] -> 45 [16,26] -> 42
418: [11,38] -> 49 [19,22] -> 41
420: [5,84] -> 89 [6,70] -> 76 [7,60] -> 67 [10,42] -> 52 [12,35] -> 47 [14,30] -> 44 [15,28] -> 43 [20,21] -> 41
425: [5,85] -> 90 [17,25] -> 42
429: [11,39] -> 50 [13,33] -> 46
430: [5,86] -> 91 [10,43] -> 53
432: [6,72] -> 78 [8,54] -> 62 [9,48] -> 57 [12,36] -> 48 [16,27] -> 43 [18,24] -> 42
434: [7,62] -> 69 [14,31] -> 45
435: [5,87] -> 92 [15,29] -> 44
440: [5,88] -> 93 [8,55] -> 63 [10,44] -> 54 [11,40] -> 51 [20,22] -> 42
441: [7,63] -> 70 [9,49] -> 58 [21,21] -> 42
442: [13,34] -> 47 [17,26] -> 43
444: [6,74] -> 80 [12,37] -> 49
448: [7,64] -> 71 [8,56] -> 64 [14,32] -> 46 [16,28] -> 44
450: [5,90] -> 95 [6,75] -> 81 [9,50] -> 59 [10,45] -> 55 [15,30] -> 45 [18,25] -> 43
455: [5,91] -> 96 [7,65] -> 72 [13,35] -> 48
456: [6,76] -> 82 [8,57] -> 65 [12,38] -> 50 [19,24] -> 43
459: [9,51] -> 60 [17,27] -> 44
460: [5,92] -> 97 [10,46] -> 56 [20,23] -> 43
462: [6,77] -> 83 [7,66] -> 73 [11,42] -> 53 [14,33] -> 47 [21,22] -> 43
464: [8,58] -> 66 [16,29] -> 45
465: [5,93] -> 98 [15,31] -> 46
468: [6,78] -> 84 [9,52] -> 61 [12,39] -> 51 [13,36] -> 49 [18,26] -> 44
470: [5,94] -> 99 [10,47] -> 57
475: [5,95] -> 100 [19,25] -> 44
476: [7,68] -> 75 [14,34] -> 48 [17,28] -> 45
480: [5,96] -> 101 [6,80] -> 86 [8,60] -> 68 [10,48] -> 58 [12,40] -> 52 [15,32] -> 47 [16,30] -> 46 [20,24] -> 44
483: [7,69] -> 76 [21,23] -> 44
484: [11,44] -> 55 [22,22] -> 44
486: [6,81] -> 87 [9,54] -> 63 [18,27] -> 45
490: [5,98] -> 103 [7,70] -> 77 [10,49] -> 59 [14,35] -> 49
492: [6,82] -> 88 [12,41] -> 53
494: [13,38] -> 51 [19,26] -> 45
495: [5,99] -> 104 [9,55] -> 64 [11,45] -> 56 [15,33] -> 48
496: [8,62] -> 70 [16,31] -> 47
500: [10,50] -> 60 [20,25] -> 45
504: [6,84] -> 90 [7,72] -> 79 [8,63] -> 71 [9,56] -> 65 [12,42] -> 54 [14,36] -> 50 [18,28] -> 46 [21,24] -> 45
506: [11,46] -> 57 [22,23] -> 45
510: [6,85] -> 91 [10,51] -> 61 [15,34] -> 49 [17,30] -> 47
512: [8,64] -> 72 [16,32] -> 48
513: [9,57] -> 66 [19,27] -> 46
516: [6,86] -> 92 [12,43] -> 55
518: [7,74] -> 81 [14,37] -> 51
520: [8,65] -> 73 [10,52] -> 62 [13,40] -> 53 [20,26] -> 46
522: [6,87] -> 93 [9,58] -> 67 [18,29] -> 47
525: [7,75] -> 82 [15,35] -> 50 [21,25] -> 46
528: [6,88] -> 94 [8,66] -> 74 [11,48] -> 59 [12,44] -> 56 [16,33] -> 49 [22,24] -> 46
532: [7,76] -> 83 [14,38] -> 52 [19,28] -> 47
539: [7,77] -> 84 [11,49] -> 60
540: [6,90] -> 96 [9,60] -> 69 [10,54] -> 64 [12,45] -> 57 [15,36] -> 51 [18,30] -> 48 [20,27] -> 47
544: [8,68] -> 76 [16,34] -> 50 [17,32] -> 49
546: [6,91] -> 97 [7,78] -> 85 [13,42] -> 55 [14,39] -> 53 [21,26] -> 47
550: [10,55] -> 65 [11,50] -> 61 [22,25] -> 47
552: [6,92] -> 98 [8,69] -> 77 [12,46] -> 58 [23,24] -> 47
558: [6,93] -> 99 [9,62] -> 71 [18,31] -> 49
560: [7,80] -> 87 [8,70] -> 78 [10,56] -> 66 [14,40] -> 54 [16,35] -> 51 [20,28] -> 48
561: [11,51] -> 62 [17,33] -> 50
564: [6,94] -> 100 [12,47] -> 59
567: [7,81] -> 88 [9,63] -> 72 [21,27] -> 48
570: [6,95] -> 101 [10,57] -> 67 [15,38] -> 53 [19,30] -> 49
572: [11,52] -> 63 [13,44] -> 57 [22,26] -> 48
574: [7,82] -> 89 [14,41] -> 55
576: [6,96] -> 102 [8,72] -> 80 [9,64] -> 73 [12,48] -> 60 [16,36] -> 52 [18,32] -> 50 [24,24] -> 48
580: [10,58] -> 68 [20,29] -> 49
585: [9,65] -> 74 [13,45] -> 58 [15,39] -> 54
588: [6,98] -> 104 [7,84] -> 91 [12,49] -> 61 [14,42] -> 56 [21,28] -> 49
592: [8,74] -> 82 [16,37] -> 53
594: [6,99] -> 105 [9,66] -> 75 [11,54] -> 65 [18,33] -> 51 [22,27] -> 49
595: [7,85] -> 92 [17,35] -> 52
598: [13,46] -> 59 [23,26] -> 49
600: [8,75] -> 83 [10,60] -> 70 [12,50] -> 62 [15,40] -> 55 [20,30] -> 50 [24,25] -> 49
602: [7,86] -> 93 [14,43] -> 57
608: [8,76] -> 84 [16,38] -> 54 [19,32] -> 51
609: [7,87] -> 94 [21,29] -> 50
612: [9,68] -> 77 [12,51] -> 63 [17,36] -> 53 [18,34] -> 52
616: [7,88] -> 95 [8,77] -> 85 [11,56] -> 67 [14,44] -> 58 [22,28] -> 50
620: [10,62] -> 72 [20,31] -> 51
621: [9,69] -> 78 [23,27] -> 50
624: [8,78] -> 86 [12,52] -> 64 [13,48] -> 61 [16,39] -> 55 [24,26] -> 50
627: [11,57] -> 68 [19,33] -> 52
630: [7,90] -> 97 [9,70] -> 79 [10,63] -> 73 [14,45] -> 59 [15,42] -> 57 [18,35] -> 53 [21,30] -> 51
637: [7,91] -> 98 [13,49] -> 62
638: [11,58] -> 69 [22,29] -> 51
640: [8,80] -> 88 [10,64] -> 74 [16,40] -> 56 [20,32] -> 52
644: [7,92] -> 99 [14,46] -> 60 [23,28] -> 51
646: [17,38] -> 55 [19,34] -> 53
648: [8,81] -> 89 [9,72] -> 81 [12,54] -> 66 [18,36] -> 54 [24,27] -> 51
650: [10,65] -> 75 [13,50] -> 63 [25,26] -> 51
651: [7,93] -> 100 [21,31] -> 52
656: [8,82] -> 90 [16,41] -> 57
658: [7,94] -> 101 [14,47] -> 61
660: [10,66] -> 76 [11,60] -> 71 [12,55] -> 67 [15,44] -> 59 [20,33] -> 53 [22,30] -> 52
663: [13,51] -> 64 [17,39] -> 56
665: [7,95] -> 102 [19,35] -> 54
666: [9,74] -> 83 [18,37] -> 55
672: [7,96] -> 103 [8,84] -> 92 [12,56] -> 68 [14,48] -> 62 [16,42] -> 58 [21,32] -> 53 [24,28] -> 52
675: [9,75] -> 84 [15,45] -> 60 [25,27] -> 52
676: [13,52] -> 65 [26,26] -> 52
680: [8,85] -> 93 [10,68] -> 78 [17,40] -> 57 [20,34] -> 54
682: [11,62] -> 73 [22,31] -> 53
684: [9,76] -> 85 [12,57] -> 69 [18,38] -> 56 [19,36] -> 55
686: [7,98] -> 105 [14,49] -> 63
688: [8,86] -> 94 [16,43] -> 59
690: [10,69] -> 79 [15,46] -> 61 [23,30] -> 53
693: [7,99] -> 106 [9,77] -> 86 [11,63] -> 74 [21,33] -> 54
696: [8,87] -> 95 [12,58] -> 70 [24,29] -> 53
700: [10,70] -> 80 [14,50] -> 64 [20,35] -> 55 [25,28] -> 53
702: [9,78] -> 87 [13,54] -> 67 [18,39] -> 57 [26,27] -> 53
704: [8,88] -> 96 [11,64] -> 75 [16,44] -> 60 [22,32] -> 54
714: [14,51] -> 65 [17,42] -> 59 [21,34] -> 55
715: [11,65] -> 76 [13,55] -> 68
720: [8,90] -> 98 [9,80] -> 89 [10,72] -> 82 [12,60] -> 72 [15,48] -> 63 [16,45] -> 61 [18,40] -> 58 [20,36] -> 56 [24,30] -> 54
726: [11,66] -> 77 [22,33] -> 55
728: [8,91] -> 99 [13,56] -> 69 [14,52] -> 66 [26,28] -> 54
729: [9,81] -> 90 [27,27] -> 54
735: [15,49] -> 64 [21,35] -> 56
736: [8,92] -> 100 [16,46] -> 62 [23,32] -> 55
738: [9,82] -> 91 [18,41] -> 59
740: [10,74] -> 84 [20,37] -> 57
741: [13,57] -> 70 [19,39] -> 58
744: [8,93] -> 101 [12,62] -> 74 [24,31] -> 55
748: [11,68] -> 79 [17,44] -> 61 [22,34] -> 56
750: [10,75] -> 85 [15,50] -> 65 [25,30] -> 55
752: [8,94] -> 102 [16,47] -> 63
754: [13,58] -> 71 [26,29] -> 55
756: [9,84] -> 93 [12,63] -> 75 [14,54] -> 68 [18,42] -> 60 [21,36] -> 57 [27,28] -> 55
759: [11,69] -> 80 [23,33] -> 56
760: [8,95] -> 103 [10,76] -> 86 [19,40] -> 59 [20,38] -> 58
765: [9,85] -> 94 [15,51] -> 66 [17,45] -> 62
768: [8,96] -> 104 [12,64] -> 76 [16,48] -> 64 [24,32] -> 56
770: [10,77] -> 87 [11,70] -> 81 [14,55] -> 69 [22,35] -> 57
774: [9,86] -> 95 [18,43] -> 61
780: [10,78] -> 88 [12,65] -> 77 [13,60] -> 73 [15,52] -> 67 [20,39] -> 59 [26,30] -> 56
782: [17,46] -> 63 [23,34] -> 57
783: [9,87] -> 96 [27,29] -> 56
784: [8,98] -> 106 [14,56] -> 70 [16,49] -> 65 [28,28] -> 56
792: [8,99] -> 107 [9,88] -> 97 [11,72] -> 83 [12,66] -> 78 [18,44] -> 62 [22,36] -> 58 [24,33] -> 57
798: [14,57] -> 71 [19,42] -> 61 [21,38] -> 59
800: [10,80] -> 90 [16,50] -> 66 [20,40] -> 60 [25,32] -> 57
806: [13,62] -> 75 [26,31] -> 57
810: [9,90] -> 99 [10,81] -> 91 [15,54] -> 69 [18,45] -> 63 [27,30] -> 57
812: [14,58] -> 72 [28,29] -> 57
814: [11,74] -> 85 [22,37] -> 59
816: [12,68] -> 80 [16,51] -> 67 [17,48] -> 65 [24,34] -> 58
819: [9,91] -> 100 [13,63] -> 76 [21,39] -> 60
820: [10,82] -> 92 [20,41] -> 61
825: [11,75] -> 86 [15,55] -> 70 [25,33] -> 58
828: [9,92] -> 101 [12,69] -> 81 [18,46] -> 64 [23,36] -> 59
832: [13,64] -> 77 [16,52] -> 68 [26,32] -> 58
836: [11,76] -> 87 [19,44] -> 63 [22,38] -> 60
837: [9,93] -> 102 [27,31] -> 58
840: [10,84] -> 94 [12,70] -> 82 [14,60] -> 74 [15,56] -> 71 [20,42] -> 62 [21,40] -> 61 [24,35] -> 59 [28,30] -> 58
846: [9,94] -> 103 [18,47] -> 65
850: [10,85] -> 95 [17,50] -> 67 [25,34] -> 59
855: [9,95] -> 104 [15,57] -> 72 [19,45] -> 64
858: [11,78] -> 89 [13,66] -> 79 [22,39] -> 61 [26,33] -> 59
860: [10,86] -> 96 [20,43] -> 63
864: [9,96] -> 105 [12,72] -> 84 [16,54] -> 70 [18,48] -> 66 [24,36] -> 60 [27,32] -> 59
868: [14,62] -> 76 [28,31] -> 59
870: [10,87] -> 97 [15,58] -> 73 [29,30] -> 59
874: [19,46] -> 65 [23,38] -> 61
880: [10,88] -> 98 [11,80] -> 91 [16,55] -> 71 [20,44] -> 64 [22,40] -> 62
882: [9,98] -> 107 [14,63] -> 77 [18,49] -> 67 [21,42] -> 63
884: [13,68] -> 81 [17,52] -> 69 [26,34] -> 60
888: [12,74] -> 86 [24,37] -> 61
891: [9,99] -> 108 [11,81] -> 92 [27,33] -> 60
896: [14,64] -> 78 [16,56] -> 72 [28,32] -> 60
897: [13,69] -> 82 [23,39] -> 62
900: [10,90] -> 100 [12,75] -> 87 [15,60] -> 75 [18,50] -> 68 [20,45] -> 65 [25,36] -> 61 [30,30] -> 60
902: [11,82] -> 93 [22,41] -> 63
910: [10,91] -> 101 [13,70] -> 83 [14,65] -> 79 [26,35] -> 61
912: [12,76] -> 88 [16,57] -> 73 [19,48] -> 67 [24,38] -> 62
918: [17,54] -> 71 [18,51] -> 69 [27,34] -> 61
920: [10,92] -> 102 [20,46] -> 66 [23,40] -> 63
924: [11,84] -> 95 [12,77] -> 89 [14,66] -> 80 [21,44] -> 65 [22,42] -> 64 [28,33] -> 61
928: [16,58] -> 74 [29,32] -> 61
930: [10,93] -> 103 [15,62] -> 77 [30,31] -> 61
935: [11,85] -> 96 [17,55] -> 72
936: [12,78] -> 90 [13,72] -> 85 [18,52] -> 70 [24,39] -> 63 [26,36] -> 62
940: [10,94] -> 104 [20,47] -> 67
945: [15,63] -> 78 [21,45] -> 66 [27,35] -> 62
946: [11,86] -> 97 [22,43] -> 65
950: [10,95] -> 105 [19,50] -> 69 [25,38] -> 63
952: [14,68] -> 82 [17,56] -> 73 [28,34] -> 62
957: [11,87] -> 98 [29,33] -> 62
960: [10,96] -> 106 [12,80] -> 92 [15,64] -> 79 [16,60] -> 76 [20,48] -> 68 [24,40] -> 64 [30,32] -> 62
962: [13,74] -> 87 [26,37] -> 63
966: [14,69] -> 83 [21,46] -> 67 [23,42] -> 65
968: [11,88] -> 99 [22,44] -> 66
969: [17,57] -> 74 [19,51] -> 70
972: [12,81] -> 93 [18,54] -> 72 [27,36] -> 63
975: [13,75] -> 88 [15,65] -> 80 [25,39] -> 64
980: [10,98] -> 108 [14,70] -> 84 [20,49] -> 69 [28,35] -> 63
984: [12,82] -> 94 [24,41] -> 65
986: [17,58] -> 75 [29,34] -> 63
988: [13,76] -> 89 [19,52] -> 71 [26,38] -> 64
990: [10,99] -> 109 [11,90] -> 101 [15,66] -> 81 [18,55] -> 73 [22,45] -> 67 [30,33] -> 63
992: [16,62] -> 78 [31,32] -> 63
1000: [20,50] -> 70 [25,40] -> 65
1001: [11,91] -> 102 [13,77] -> 90
1008: [12,84] -> 96 [14,72] -> 86 [16,63] -> 79 [18,56] -> 74 [21,48] -> 69 [24,42] -> 66 [28,36] -> 64
1012: [11,92] -> 103 [22,46] -> 68 [23,44] -> 67
1014: [13,78] -> 91 [26,39] -> 65
1020: [12,85] -> 97 [15,68] -> 83 [17,60] -> 77 [20,51] -> 71 [30,34] -> 64
1023: [11,93] -> 104 [31,33] -> 64
1024: [16,64] -> 80 [32,32] -> 64
1026: [18,57] -> 75 [19,54] -> 73 [27,38] -> 65
1032: [12,86] -> 98 [24,43] -> 67
1034: [11,94] -> 105 [22,47] -> 69
1035: [15,69] -> 84 [23,45] -> 68
1036: [14,74] -> 88 [28,37] -> 65
1040: [13,80] -> 93 [16,65] -> 81 [20,52] -> 72 [26,40] -> 66
1044: [12,87] -> 99 [18,58] -> 76 [29,36] -> 65
1045: [11,95] -> 106 [19,55] -> 74
1050: [14,75] -> 89 [15,70] -> 85 [21,50] -> 71 [25,42] -> 67 [30,35] -> 65
1053: [13,81] -> 94 [27,39] -> 66
1054: [17,62] -> 79 [31,34] -> 65
1056: [11,96] -> 107 [12,88] -> 100 [16,66] -> 82 [22,48] -> 70 [24,44] -> 68 [32,33] -> 65
1064: [14,76] -> 90 [19,56] -> 75 [28,38] -> 66
1066: [13,82] -> 95 [26,41] -> 67
1071: [17,63] -> 80 [21,51] -> 72
1078: [11,98] -> 109 [14,77] -> 91 [22,49] -> 71
1080: [12,90] -> 102 [15,72] -> 87 [18,60] -> 78 [20,54] -> 74 [24,45] -> 69 [27,40] -> 67 [30,36] -> 66
1088: [16,68] -> 84 [17,64] -> 81 [32,34] -> 66
1089: [11,99] -> 110 [33,33] -> 66
1092: [12,91] -> 103 [13,84] -> 97 [14,78] -> 92 [21,52] -> 73 [26,42] -> 68 [28,39] -> 67
1100: [20,55] -> 75 [22,50] -> 72 [25,44] -> 69
1102: [19,58] -> 77 [29,38] -> 67
1104: [12,92] -> 104 [16,69] -> 85 [23,48] -> 71 [24,46] -> 70
1105: [13,85] -> 98 [17,65] -> 82
1110: [15,74] -> 89 [30,37] -> 67
1116: [12,93] -> 105 [18,62] -> 80 [31,36] -> 67
1118: [13,86] -> 99 [26,43] -> 69
1120: [14,80] -> 94 [16,70] -> 86 [20,56] -> 76 [28,40] -> 68 [32,35] -> 67
1122: [17,66] -> 83 [22,51] -> 73 [33,34] -> 67
1125: [15,75] -> 90 [25,45] -> 70
1128: [12,94] -> 106 [24,47] -> 71
1131: [13,87] -> 100 [29,39] -> 68
1134: [14,81] -> 95 [18,63] -> 81 [21,54] -> 75 [27,42] -> 69
1140: [12,95] -> 107 [15,76] -> 91 [19,60] -> 79 [20,57] -> 77 [30,38] -> 68
1144: [13,88] -> 101 [22,52] -> 74 [26,44] -> 70
1148: [14,82] -> 96 [28,41] -> 69
1150: [23,50] -> 73 [25,46] -> 71
1152: [12,96] -> 108 [16,72] -> 88 [18,64] -> 82 [24,48] -> 72 [32,36] -> 68
1155: [15,77] -> 92 [21,55] -> 76 [33,35] -> 68
1156: [17,68] -> 85 [34,34] -> 68
1160: [20,58] -> 78 [29,40] -> 69
1170: [13,90] -> 103 [15,78] -> 93 [18,65] -> 83 [26,45] -> 71 [30,39] -> 69
1173: [17,69] -> 86 [23,51] -> 74
1176: [12,98] -> 110 [14,84] -> 98 [21,56] -> 77 [24,49] -> 73 [28,42] -> 70
1178: [19,62] -> 81 [31,38] -> 69
1184: [16,74] -> 90 [32,37] -> 69
1188: [12,99] -> 111 [18,66] -> 84 [22,54] -> 76 [27,44] -> 71 [33,36] -> 69
1190: [14,85] -> 99 [17,70] -> 87 [34,35] -> 69
1196: [13,92] -> 105 [23,52] -> 75 [26,46] -> 72
1197: [19,63] -> 82 [21,57] -> 78
1200: [15,80] -> 95 [16,75] -> 91 [20,60] -> 80 [24,50] -> 74 [25,48] -> 73 [30,40] -> 70
1204: [14,86] -> 100 [28,43] -> 71
1209: [13,93] -> 106 [31,39] -> 70
1215: [15,81] -> 96 [27,45] -> 72
1216: [16,76] -> 92 [19,64] -> 83 [32,38] -> 70
1218: [14,87] -> 101 [21,58] -> 79 [29,42] -> 71
1222: [13,94] -> 107 [26,47] -> 73
1224: [17,72] -> 89 [18,68] -> 86 [24,51] -> 75 [34,36] -> 70
1225: [25,49] -> 74 [35,35] -> 70
1230: [15,82] -> 97 [30,41] -> 71
1232: [14,88] -> 102 [16,77] -> 93 [22,56] -> 78 [28,44] -> 72
1235: [13,95] -> 108 [19,65] -> 84
1240: [20,62] -> 82 [31,40] -> 71
1242: [18,69] -> 87 [23,54] -> 77 [27,46] -> 73
1248: [13,96] -> 109 [16,78] -> 94 [24,52] -> 76 [26,48] -> 74 [32,39] -> 71
1254: [19,66] -> 85 [22,57] -> 79 [33,38] -> 71
1258: [17,74] -> 91 [34,37] -> 71
1260: [14,90] -> 104 [15,84] -> 99 [18,70] -> 88 [20,63] -> 83 [21,60] -> 81 [28,45] -> 73 [30,42] -> 72 [35,36] -> 71
1274: [13,98] -> 111 [14,91] -> 105 [26,49] -> 75
1275: [15,85] -> 100 [17,75] -> 92 [25,51] -> 76
1276: [22,58] -> 80 [29,44] -> 73
1280: [16,80] -> 96 [20,64] -> 84 [32,40] -> 72
1287: [13,99] -> 112 [33,39] -> 72
1288: [14,92] -> 106 [23,56] -> 79 [28,46] -> 74
1290: [15,86] -> 101 [30,43] -> 73
1292: [17,76] -> 93 [19,68] -> 87 [34,38] -> 72
1296: [16,81] -> 97 [18,72] -> 90 [24,54] -> 78 [27,48] -> 75 [36,36] -> 72
1300: [20,65] -> 85 [25,52] -> 77 [26,50] -> 76
1302: [14,93] -> 107 [21,62] -> 83 [31,42] -> 73
1305: [15,87] -> 102 [29,45] -> 74
1311: [19,69] -> 88 [23,57] -> 80
1312: [16,82] -> 98 [32,41] -> 73
1316: [14,94] -> 108 [28,47] -> 75
1320: [15,88] -> 103 [20,66] -> 86 [22,60] -> 82 [24,55] -> 79 [30,44] -> 74 [33,40] -> 73
1323: [21,63] -> 84 [27,49] -> 76
1326: [17,78] -> 95 [26,51] -> 77 [34,39] -> 73
1330: [14,95] -> 109 [19,70] -> 89 [35,38] -> 73
1332: [18,74] -> 92 [36,37] -> 73
1334: [23,58] -> 81 [29,46] -> 75
1344: [14,96] -> 110 [16,84] -> 100 [21,64] -> 85 [24,56] -> 80 [28,48] -> 76 [32,42] -> 74
1350: [15,90] -> 105 [18,75] -> 93 [25,54] -> 79 [27,50] -> 77 [30,45] -> 75
1360: [16,85] -> 101 [17,80] -> 97 [20,68] -> 88 [34,40] -> 74
1364: [22,62] -> 84 [31,44] -> 75
1365: [15,91] -> 106 [21,65] -> 86 [35,39] -> 74
1368: [18,76] -> 94 [19,72] -> 91 [24,57] -> 81 [36,38] -> 74
1372: [14,98] -> 112 [28,49] -> 77
1376: [16,86] -> 102 [32,43] -> 75
1377: [17,81] -> 98 [27,51] -> 78
1380: [15,92] -> 107 [20,69] -> 89 [23,60] -> 83 [30,46] -> 76
1386: [14,99] -> 113 [18,77] -> 95 [21,66] -> 87 [22,63] -> 85 [33,42] -> 75
1392: [16,87] -> 103 [24,58] -> 82 [29,48] -> 77
1394: [17,82] -> 99 [34,41] -> 75
1395: [15,93] -> 108 [31,45] -> 76
1400: [20,70] -> 90 [25,56] -> 81 [28,50] -> 78 [35,40] -> 75
1404: [18,78] -> 96 [26,54] -> 80 [27,52] -> 79 [36,39] -> 75
1406: [19,74] -> 93 [37,38] -> 75
1408: [16,88] -> 104 [22,64] -> 86 [32,44] -> 76
1410: [15,94] -> 109 [30,47] -> 77
1425: [15,95] -> 110 [19,75] -> 94 [25,57] -> 82
1426: [23,62] -> 85 [31,46] -> 77
1428: [17,84] -> 101 [21,68] -> 89 [28,51] -> 79 [34,42] -> 76
1430: [22,65] -> 87 [26,55] -> 81
1440: [15,96] -> 111 [16,90] -> 106 [18,80] -> 98 [20,72] -> 92 [24,60] -> 84 [30,48] -> 78 [32,45] -> 77 [36,40] -> 76
1444: [19,76] -> 95 [38,38] -> 76
1449: [21,69] -> 90 [23,63] -> 86
1450: [25,58] -> 83 [29,50] -> 79
1452: [22,66] -> 88 [33,44] -> 77
1456: [16,91] -> 107 [26,56] -> 82 [28,52] -> 80
1458: [18,81] -> 99 [27,54] -> 81
1462: [17,86] -> 103 [34,43] -> 77
1470: [15,98] -> 113 [21,70] -> 91 [30,49] -> 79 [35,42] -> 77
1472: [16,92] -> 108 [23,64] -> 87 [32,46] -> 78
1476: [18,82] -> 100 [36,41] -> 77
1479: [17,87] -> 104 [29,51] -> 80
1480: [20,74] -> 94 [37,40] -> 77
1482: [19,78] -> 97 [26,57] -> 83 [38,39] -> 77
1485: [15,99] -> 114 [27,55] -> 82 [33,45] -> 78
1488: [16,93] -> 109 [24,62] -> 86 [31,48] -> 79
1496: [17,88] -> 105 [22,68] -> 90 [34,44] -> 78
1500: [20,75] -> 95 [25,60] -> 85 [30,50] -> 80
1504: [16,94] -> 110 [32,47] -> 79
1508: [26,58] -> 84 [29,52] -> 81
1512: [18,84] -> 102 [21,72] -> 93 [24,63] -> 87 [27,56] -> 83 [28,54] -> 82 [36,42] -> 78
1518: [22,69] -> 91 [23,66] -> 89 [33,46] -> 79
1520: [16,95] -> 111 [19,80] -> 99 [20,76] -> 96 [38,40] -> 78
1530: [17,90] -> 107 [18,85] -> 103 [30,51] -> 81 [34,45] -> 79
1536: [16,96] -> 112 [24,64] -> 88 [32,48] -> 80
1539: [19,81] -> 100 [27,57] -> 84
1540: [20,77] -> 97 [22,70] -> 92 [28,55] -> 83 [35,44] -> 79
1548: [18,86] -> 104 [36,43] -> 79
1550: [25,62] -> 87 [31,50] -> 81
1554: [21,74] -> 95 [37,42] -> 79
1558: [19,82] -> 101 [38,41] -> 79
1560: [20,78] -> 98 [24,65] -> 89 [26,60] -> 86 [30,52] -> 82 [39,40] -> 79
1564: [17,92] -> 109 [23,68] -> 91 [34,46] -> 80
1566: [18,87] -> 105 [27,58] -> 85 [29,54] -> 83
1568: [16,98] -> 114 [28,56] -> 84 [32,49] -> 81
1575: [21,75] -> 96 [25,63] -> 88 [35,45] -> 80
1581: [17,93] -> 110 [31,51] -> 82
1584: [16,99] -> 115 [18,88] -> 106 [22,72] -> 94 [24,66] -> 90 [33,48] -> 81 [36,44] -> 80
1596: [19,84] -> 103 [21,76] -> 97 [28,57] -> 85 [38,42] -> 80
1598: [17,94] -> 111 [34,47] -> 81
1600: [20,80] -> 100 [25,64] -> 89 [32,50] -> 82 [40,40] -> 80
1610: [23,70] -> 93 [35,46] -> 81
1612: [26,62] -> 88 [31,52] -> 83
1615: [17,95] -> 112 [19,85] -> 104
1617: [21,77] -> 98 [33,49] -> 82
1620: [18,90] -> 108 [20,81] -> 101 [27,60] -> 87 [30,54] -> 84 [36,45] -> 81
1624: [28,58] -> 86 [29,56] -> 85
1628: [22,74] -> 96 [37,44] -> 81
1632: [17,96] -> 113 [24,68] -> 92 [32,51] -> 83 [34,48] -> 82
1634: [19,86] -> 105 [38,43] -> 81
1638: [18,91] -> 109 [21,78] -> 99 [26,63] -> 89 [39,42] -> 81
1640: [20,82] -> 102 [40,41] -> 81
1650: [22,75] -> 97 [25,66] -> 91 [30,55] -> 85 [33,50] -> 83
1653: [19,87] -> 106 [29,57] -> 86
1656: [18,92] -> 110 [23,72] -> 95 [24,69] -> 93 [36,46] -> 82
1664: [26,64] -> 90 [32,52] -> 84
1666: [17,98] -> 115 [34,49] -> 83
1672: [19,88] -> 107 [22,76] -> 98 [38,44] -> 82
1674: [18,93] -> 111 [27,62] -> 89 [31,54] -> 85
1680: [20,84] -> 104 [21,80] -> 101 [24,70] -> 94 [28,60] -> 88 [30,56] -> 86 [35,48] -> 83 [40,42] -> 82
1683: [17,99] -> 116 [33,51] -> 84
1692: [18,94] -> 112 [36,47] -> 83
1700: [20,85] -> 105 [25,68] -> 93 [34,50] -> 84
1701: [21,81] -> 102 [27,63] -> 90
1702: [23,74] -> 97 [37,46] -> 83
1710: [18,95] -> 113 [19,90] -> 109 [30,57] -> 87 [38,45] -> 83
1716: [22,78] -> 100 [26,66] -> 92 [33,52] -> 85 [39,44] -> 83
1720: [20,86] -> 106 [40,43] -> 83
1722: [21,82] -> 103 [41,42] -> 83
1725: [23,75] -> 98 [25,69] -> 94
1728: [18,96] -> 114 [24,72] -> 96 [27,64] -> 91 [32,54] -> 86 [36,48] -> 84
1736: [28,62] -> 90 [31,56] -> 87
1740: [20,87] -> 107 [29,60] -> 89 [30,58] -> 88
1748: [19,92] -> 111 [23,76] -> 99 [38,46] -> 84
1750: [25,70] -> 95 [35,50] -> 85
1755: [27,65] -> 92 [39,45] -> 84
1760: [20,88] -> 108 [22,80] -> 102 [32,55] -> 87 [40,44] -> 84
1764: [18,98] -> 116 [21,84] -> 105 [28,63] -> 91 [36,49] -> 85 [42,42] -> 84
1767: [19,93] -> 112 [31,57] -> 88
1768: [26,68] -> 94 [34,52] -> 86
1776: [24,74] -> 98 [37,48] -> 85
1782: [18,99] -> 117 [22,81] -> 103 [27,66] -> 93 [33,54] -> 87
1785: [21,85] -> 106 [35,51] -> 86
1786: [19,94] -> 113 [38,47] -> 85
1792: [28,64] -> 92 [32,56] -> 88
1794: [23,78] -> 101 [26,69] -> 95 [39,46] -> 85
1798: [29,62] -> 91 [31,58] -> 89
1800: [20,90] -> 110 [24,75] -> 99 [25,72] -> 97 [30,60] -> 90 [36,50] -> 86 [40,45] -> 85
1804: [22,82] -> 104 [41,44] -> 85
1806: [21,86] -> 107 [42,43] -> 85
1820: [20,91] -> 111 [26,70] -> 96 [28,65] -> 93 [35,52] -> 87
1824: [19,96] -> 115 [24,76] -> 100 [32,57] -> 89 [38,48] -> 86
1827: [21,87] -> 108 [29,63] -> 92
1836: [27,68] -> 95 [34,54] -> 88 [36,51] -> 87
1840: [20,92] -> 112 [23,80] -> 103 [40,46] -> 86
1848: [21,88] -> 109 [22,84] -> 106 [24,77] -> 101 [28,66] -> 94 [33,56] -> 89 [42,44] -> 86
1850: [25,74] -> 99 [37,50] -> 87
1856: [29,64] -> 93 [32,58] -> 90
1860: [20,93] -> 113 [30,62] -> 92 [31,60] -> 91
1862: [19,98] -> 117 [38,49] -> 87
1863: [23,81] -> 104 [27,69] -> 96
1870: [22,85] -> 107 [34,55] -> 89
1872: [24,78] -> 102 [26,72] -> 98 [36,52] -> 88 [39,48] -> 87
1880: [20,94] -> 114 [40,47] -> 87
1881: [19,99] -> 118 [33,57] -> 90
1886: [23,82] -> 105 [41,46] -> 87
1890: [21,90] -> 111 [27,70] -> 97 [30,63] -> 93 [35,54] -> 89 [42,45] -> 87
1892: [22,86] -> 108 [43,44] -> 87
1900: [20,95] -> 115 [25,76] -> 101 [38,50] -> 88
1904: [28,68] -> 96 [34,56] -> 90
1911: [21,91] -> 112 [39,49] -> 88
1914: [22,87] -> 109 [29,66] -> 95 [33,58] -> 91
1920: [20,96] -> 116 [24,80] -> 104 [30,64] -> 94 [32,60] -> 92 [40,48] -> 88
1924: [26,74] -> 100 [37,52] -> 89
1925: [25,77] -> 102 [35,55] -> 90
1932: [21,92] -> 113 [23,84] -> 107 [28,69] -> 97 [42,46] -> 88
1936: [22,88] -> 110 [44,44] -> 88
1938: [34,57] -> 91 [38,51] -> 89
1944: [24,81] -> 105 [27,72] -> 99 [36,54] -> 90
1950: [25,78] -> 103 [26,75] -> 101 [30,65] -> 95 [39,50] -> 89
1953: [21,93] -> 114 [31,63] -> 94
1960: [20,98] -> 118 [28,70] -> 98 [35,56] -> 91 [40,49] -> 89
1968: [24,82] -> 106 [41,48] -> 89
1972: [29,68] -> 97 [34,58] -> 92
1974: [21,94] -> 115 [42,47] -> 89
1976: [26,76] -> 102 [38,52] -> 90
1978: [23,86] -> 109 [43,46] -> 89
1980: [20,99] -> 119 [22,90] -> 112 [30,66] -> 96 [33,60] -> 93 [36,55] -> 91 [44,45] -> 89
1984: [31,64] -> 95 [32,62] -> 94
1995: [21,95] -> 116 [35,57] -> 92
1998: [27,74] -> 101 [37,54] -> 91
2000: [25,80] -> 105 [40,50] -> 90
2001: [23,87] -> 110 [29,69] -> 98
2002: [22,91] -> 113 [26,77] -> 103
2016: [21,96] -> 117 [24,84] -> 108 [28,72] -> 100 [32,63] -> 95 [36,56] -> 92 [42,48] -> 90
2024: [22,92] -> 114 [23,88] -> 111 [44,46] -> 90
2025: [25,81] -> 106 [27,75] -> 102 [45,45] -> 90
2028: [26,78] -> 104 [39,52] -> 91
2030: [29,70] -> 99 [35,58] -> 93
2040: [24,85] -> 109 [30,68] -> 98 [34,60] -> 94 [40,51] -> 91
2046: [22,93] -> 115 [31,66] -> 97 [33,62] -> 95
2050: [25,82] -> 107 [41,50] -> 91
2052: [27,76] -> 103 [36,57] -> 93 [38,54] -> 92
2058: [21,98] -> 119 [42,49] -> 91
2064: [24,86] -> 110 [43,48] -> 91
2068: [22,94] -> 116 [44,47] -> 91
2070: [23,90] -> 113 [30,69] -> 99 [45,46] -> 91
2072: [28,74] -> 102 [37,56] -> 93
2079: [21,99] -> 120 [27,77] -> 104 [33,63] -> 96
2080: [26,80] -> 106 [32,65] -> 97 [40,52] -> 92
2088: [24,87] -> 111 [29,72] -> 101 [36,58] -> 94
2090: [22,95] -> 117 [38,55] -> 93
2100: [25,84] -> 109 [28,75] -> 103 [30,70] -> 100 [35,60] -> 95 [42,50] -> 92
2106: [26,81] -> 107 [27,78] -> 105 [39,54] -> 93
2108: [31,68] -> 99 [34,62] -> 96
2112: [22,96] -> 118 [24,88] -> 112 [32,66] -> 98 [33,64] -> 97 [44,48] -> 92
2116: [23,92] -> 115 [46,46] -> 92
2128: [28,76] -> 104 [38,56] -> 94
2132: [26,82] -> 108 [41,52] -> 93
2139: [23,93] -> 116 [31,69] -> 100
2142: [34,63] -> 97 [42,51] -> 93
2145: [33,65] -> 98 [39,55] -> 94
2146: [29,74] -> 103 [37,58] -> 95
2150: [25,86] -> 111 [43,50] -> 93
2156: [22,98] -> 120 [28,77] -> 105 [44,49] -> 93
2160: [24,90] -> 114 [27,80] -> 107 [30,72] -> 102 [36,60] -> 96 [40,54] -> 94 [45,48] -> 93
2162: [23,94] -> 117 [46,47] -> 93
2170: [31,70] -> 101 [35,62] -> 97
2175: [25,87] -> 112 [29,75] -> 104
2176: [32,68] -> 100 [34,64] -> 98
2178: [22,99] -> 121 [33,66] -> 99
2184: [24,91] -> 115 [26,84] -> 110 [28,78] -> 106 [39,56] -> 95 [42,52] -> 94
2200: [25,88] -> 113 [40,55] -> 95 [44,50] -> 94
2204: [29,76] -> 105 [38,58] -> 96
2205: [35,63] -> 98 [45,49] -> 94
2208: [23,96] -> 119 [24,92] -> 116 [32,69] -> 101 [46,48] -> 94
2210: [26,85] -> 111 [34,65] -> 99
2214: [27,82] -> 109 [41,54] -> 95
2220: [30,74] -> 104 [37,60] -> 97
2232: [24,93] -> 117 [31,72] -> 103 [36,62] -> 98
2236: [26,86] -> 112 [43,52] -> 95
2240: [28,80] -> 108 [32,70] -> 102 [35,64] -> 99 [40,56] -> 96
2244: [33,68] -> 101 [34,66] -> 100 [44,51] -> 95
2250: [25,90] -> 115 [30,75] -> 105 [45,50] -> 95
2254: [23,98] -> 121 [46,49] -> 95
2256: [24,94] -> 118 [47,48] -> 95
2262: [26,87] -> 113 [29,78] -> 107 [39,58] -> 97
2268: [27,84] -> 111 [28,81] -> 109 [36,63] -> 99 [42,54] -> 96
2275: [25,91] -> 116 [35,65] -> 100
2277: [23,99] -> 122 [33,69] -> 102
2280: [24,95] -> 119 [30,76] -> 106 [38,60] -> 98 [40,57] -> 97
2288: [26,88] -> 114 [44,52] -> 96
2294: [31,74] -> 105 [37,62] -> 99
2295: [27,85] -> 112 [45,51] -> 96
2296: [28,82] -> 110 [41,56] -> 97
2300: [25,92] -> 117 [46,50] -> 96
2304: [24,96] -> 120 [32,72] -> 104 [36,64] -> 100 [48,48] -> 96
2310: [30,77] -> 107 [33,70] -> 103 [35,66] -> 101 [42,55] -> 97
2320: [29,80] -> 109 [40,58] -> 98
2322: [27,86] -> 113 [43,54] -> 97
2325: [25,93] -> 118 [31,75] -> 106
2340: [26,90] -> 116 [30,78] -> 108 [36,65] -> 101 [39,60] -> 99 [45,52] -> 97
2346: [34,69] -> 103 [46,51] -> 97
2349: [27,87] -> 114 [29,81] -> 110
2350: [25,94] -> 119 [47,50] -> 97
2352: [24,98] -> 122 [28,84] -> 112 [42,56] -> 98 [48,49] -> 97
2356: [31,76] -> 107 [38,62] -> 100
2368: [32,74] -> 106 [37,64] -> 101
2376: [24,99] -> 123 [27,88] -> 115 [33,72] -> 105 [36,66] -> 102 [44,54] -> 98
2378: [29,82] -> 111 [41,58] -> 99
2380: [28,85] -> 113 [34,70] -> 104 [35,68] -> 103
2392: [26,92] -> 118 [46,52] -> 98
2394: [38,63] -> 101 [42,57] -> 99
2400: [25,96] -> 121 [30,80] -> 110 [32,75] -> 107 [40,60] -> 100 [48,50] -> 98
2408: [28,86] -> 114 [43,56] -> 99
2418: [26,93] -> 119 [31,78] -> 109 [39,62] -> 101
2430: [27,90] -> 117 [30,81] -> 111 [45,54] -> 99
2432: [32,76] -> 108 [38,64] -> 102
2436: [28,87] -> 115 [29,84] -> 113 [42,58] -> 100
2442: [33,74] -> 107 [37,66] -> 103
2444: [26,94] -> 120 [47,52] -> 99
2448: [34,72] -> 106 [36,68] -> 104 [48,51] -> 99
2450: [25,98] -> 123 [35,70] -> 105 [49,50] -> 99
2457: [27,91] -> 118 [39,63] -> 102
2460: [30,82] -> 112 [41,60] -> 101
2464: [28,88] -> 116 [32,77] -> 109 [44,56] -> 100
2470: [26,95] -> 121 [38,65] -> 103
2475: [25,99] -> 124 [33,75] -> 108 [45,55] -> 100
2480: [31,80] -> 111 [40,62] -> 102
2484: [27,92] -> 119 [36,69] -> 105 [46,54] -> 100
2494: [29,86] -> 115 [43,58] -> 101
2496: [26,96] -> 122 [32,78] -> 110 [39,64] -> 103 [48,52] -> 100
2508: [33,76] -> 109 [38,66] -> 104 [44,57] -> 101
2511: [27,93] -> 120 [31,81] -> 112
2516: [34,74] -> 108 [37,68] -> 105
2520: [28,90] -> 118 [30,84] -> 114 [35,72] -> 107 [36,70] -> 106 [40,63] -> 103 [42,60] -> 102 [45,56] -> 101
2538: [27,94] -> 121 [47,54] -> 101
2542: [31,82] -> 113 [41,62] -> 103
2548: [26,98] -> 124 [28,91] -> 119 [49,52] -> 101
2550: [30,85] -> 115 [34,75] -> 109 [50,51] -> 101
2552: [29,88] -> 117 [44,58] -> 102
2560: [32,80] -> 112 [40,64] -> 104
2565: [27,95] -> 122 [45,57] -> 102
2574: [26,99] -> 125 [33,78] -> 111 [39,66] -> 105
2576: [28,92] -> 120 [46,56] -> 102
2580: [30,86] -> 116 [43,60] -> 103
2584: [34,76] -> 110 [38,68] -> 106
2590: [35,74] -> 109 [37,70] -> 107
2592: [27,96] -> 123 [32,81] -> 113 [36,72] -> 108 [48,54] -> 102
2600: [40,65] -> 105 [50,52] -> 102
2604: [28,93] -> 121 [31,84] -> 115 [42,62] -> 104
2610: [29,90] -> 119 [30,87] -> 117 [45,58] -> 103
2622: [38,69] -> 107 [46,57] -> 103
2624: [32,82] -> 114 [41,64] -> 105
2632: [28,94] -> 122 [47,56] -> 103
2640: [30,88] -> 118 [33,80] -> 113 [40,66] -> 106 [44,60] -> 104 [48,55] -> 103
2646: [27,98] -> 125 [42,63] -> 105 [49,54] -> 103
2652: [34,78] -> 112 [39,68] -> 107 [51,52] -> 103
2660: [28,95] -> 123 [35,76] -> 111 [38,70] -> 108
2664: [36,74] -> 110 [37,72] -> 109
2666: [31,86] -> 117 [43,62] -> 105
2668: [29,92] -> 121 [46,58] -> 104
2673: [27,99] -> 126 [33,81] -> 114
2688: [28,96] -> 124 [32,84] -> 116 [42,64] -> 106 [48,56] -> 104
2695: [35,77] -> 112 [49,55] -> 104
2697: [29,93] -> 122 [31,87] -> 118
2700: [30,90] -> 120 [36,75] -> 111 [45,60] -> 105 [50,54] -> 104
2706: [33,82] -> 115 [41,66] -> 107
2720: [32,85] -> 117 [34,80] -> 114 [40,68] -> 108
2726: [29,94] -> 123 [47,58] -> 105
2728: [31,88] -> 119 [44,62] -> 106
2730: [30,91] -> 121 [35,78] -> 113 [39,70] -> 109 [42,65] -> 107
2736: [36,76] -> 112 [38,72] -> 110 [48,57] -> 105
2744: [28,98] -> 126 [49,56] -> 105
2752: [32,86] -> 118 [43,64] -> 107
2754: [34,81] -> 115 [51,54] -> 105
2760: [30,92] -> 122 [40,69] -> 109 [46,60] -> 106
2772: [28,99] -> 127 [33,84] -> 117 [36,77] -> 113 [42,66] -> 108 [44,63] -> 107
2784: [29,96] -> 125 [32,87] -> 119 [48,58] -> 106
2788: [34,82] -> 116 [41,68] -> 109
2790: [30,93] -> 123 [31,90] -> 121 [45,62] -> 107
2800: [35,80] -> 115 [40,70] -> 110 [50,56] -> 106
2805: [33,85] -> 118 [51,55] -> 106
2808: [36,78] -> 114 [39,72] -> 111 [52,54] -> 106
2812: [37,76] -> 113 [38,74] -> 112
2816: [32,88] -> 120 [44,64] -> 108
2820: [30,94] -> 124 [47,60] -> 107
2835: [35,81] -> 116 [45,63] -> 108
2838: [33,86] -> 119 [43,66] -> 109
2842: [29,98] -> 127 [49,58] -> 107
2850: [30,95] -> 125 [38,75] -> 113 [50,57] -> 107
2852: [31,92] -> 123 [46,62] -> 108
2856: [34,84] -> 118 [42,68] -> 110 [51,56] -> 107
2860: [44,65] -> 109 [52,55] -> 107
2870: [35,82] -> 117 [41,70] -> 111
2871: [29,99] -> 128 [33,87] -> 120
2880: [30,96] -> 126 [32,90] -> 122 [36,80] -> 116 [40,72] -> 112 [45,64] -> 109 [48,60] -> 108
2886: [37,78] -> 115 [39,74] -> 113
2898: [42,69] -> 111 [46,63] -> 109
2904: [33,88] -> 121 [44,66] -> 110
2912: [32,91] -> 123 [52,56] -> 108
2914: [31,94] -> 125 [47,62] -> 109
2916: [36,81] -> 117 [54,54] -> 108
2924: [34,86] -> 120 [43,68] -> 111
2925: [39,75] -> 114 [45,65] -> 110
2940: [30,98] -> 128 [35,84] -> 119 [42,70] -> 112 [49,60] -> 109
2944: [32,92] -> 124 [46,64] -> 110
2952: [36,82] -> 118 [41,72] -> 113
2958: [34,87] -> 121 [51,58] -> 109
2960: [37,80] -> 117 [40,74] -> 114
2964: [38,78] -> 116 [39,76] -> 115 [52,57] -> 109
2970: [30,99] -> 129 [33,90] -> 123 [45,66] -> 111 [54,55] -> 109
2976: [31,96] -> 127 [32,93] -> 125 [48,62] -> 110
2992: [34,88] -> 122 [44,68] -> 112
3000: [40,75] -> 115 [50,60] -> 110
3003: [33,91] -> 124 [39,77] -> 116
3008: [32,94] -> 126 [47,64] -> 111
3010: [35,86] -> 121 [43,70] -> 113
3024: [36,84] -> 120 [42,72] -> 114 [48,63] -> 111 [54,56] -> 110
3034: [37,82] -> 119 [41,74] -> 115
3036: [33,92] -> 125 [44,69] -> 113 [46,66] -> 112
3038: [31,98] -> 129 [49,62] -> 111
3040: [32,95] -> 127 [38,80] -> 118 [40,76] -> 116
3060: [34,90] -> 124 [36,85] -> 121 [45,68] -> 113 [51,60] -> 111
3069: [31,99] -> 130 [33,93] -> 126
3072: [32,96] -> 128 [48,64] -> 112
3078: [38,81] -> 119 [54,57] -> 111
3080: [35,88] -> 123 [40,77] -> 117 [44,70] -> 114 [55,56] -> 111
3096: [36,86] -> 122 [43,72] -> 115
3102: [33,94] -> 127 [47,66] -> 113
3108: [37,84] -> 121 [42,74] -> 116
3116: [38,82] -> 120 [41,76] -> 117
3120: [39,80] -> 119 [40,78] -> 118 [48,65] -> 113 [52,60] -> 112
3128: [34,92] -> 126 [46,68] -> 114
3132: [36,87] -> 123 [54,58] -> 112
3135: [33,95] -> 128 [55,57] -> 112
3136: [32,98] -> 130 [49,64] -> 113 [56,56] -> 112
3150: [35,90] -> 125 [42,75] -> 117 [45,70] -> 115 [50,63] -> 113
3162: [34,93] -> 127 [51,62] -> 113
3168: [32,99] -> 131 [33,96] -> 129 [36,88] -> 124 [44,72] -> 116 [48,66] -> 114
3182: [37,86] -> 123 [43,74] -> 117
3185: [35,91] -> 126 [49,65] -> 114
3192: [38,84] -> 122 [42,76] -> 118 [56,57] -> 113
3196: [34,94] -> 128 [47,68] -> 115
3198: [39,82] -> 121 [41,78] -> 119
3200: [40,80] -> 120 [50,64] -> 114
3220: [35,92] -> 127 [46,70] -> 116
3230: [34,95] -> 129 [38,85] -> 123
3234: [33,98] -> 131 [42,77] -> 119 [49,66] -> 115
3240: [36,90] -> 126 [40,81] -> 121 [45,72] -> 117 [54,60] -> 114
3256: [37,88] -> 125 [44,74] -> 118
3264: [34,96] -> 130 [48,68] -> 116 [51,64] -> 115
3268: [38,86] -> 124 [43,76] -> 119
3276: [36,91] -> 127 [39,84] -> 123 [42,78] -> 120 [52,63] -> 115
3280: [40,82] -> 122 [41,80] -> 121
3290: [35,94] -> 129 [47,70] -> 117
3300: [44,75] -> 119 [50,66] -> 116 [55,60] -> 115
3306: [38,87] -> 125 [57,58] -> 115
3312: [36,92] -> 128 [46,72] -> 118 [48,69] -> 117
3315: [39,85] -> 124 [51,65] -> 116
3330: [37,90] -> 127 [45,74] -> 119
3332: [34,98] -> 132 [49,68] -> 117
3344: [38,88] -> 126 [44,76] -> 120
3348: [36,93] -> 129 [54,62] -> 116
3354: [39,86] -> 125 [43,78] -> 121
3360: [35,96] -> 131 [40,84] -> 124 [42,80] -> 122 [48,70] -> 118 [56,60] -> 116
3366: [34,99] -> 133 [51,66] -> 117
3384: [36,94] -> 130 [47,72] -> 119
3400: [40,85] -> 125 [50,68] -> 118
3402: [42,81] -> 123 [54,63] -> 117
3404: [37,92] -> 129 [46,74] -> 120
3420: [36,95] -> 131 [38,90] -> 128 [45,76] -> 121 [57,60] -> 117
3430: [35,98] -> 133 [49,70] -> 119
3432: [39,88] -> 127 [44,78] -> 122 [52,66] -> 118
3440: [40,86] -> 126 [43,80] -> 123
3444: [41,84] -> 125 [42,82] -> 124
3450: [46,75] -> 121 [50,69] -> 119
3456: [36,96] -> 132 [48,72] -> 120 [54,64] -> 118
3465: [35,99] -> 134 [45,77] -> 122 [55,63] -> 118
3478: [37,94] -> 131 [47,74] -> 121
3480: [40,87] -> 127 [58,60] -> 118
3496: [38,92] -> 130 [46,76] -> 122
3510: [39,90] -> 129 [45,78] -> 123 [54,65] -> 119
3520: [40,88] -> 128 [44,80] -> 124 [55,64] -> 119
3526: [41,86] -> 127 [43,82] -> 125
3528: [36,98] -> 134 [42,84] -> 126 [49,72] -> 121 [56,63] -> 119
3534: [38,93] -> 131 [57,62] -> 119
3552: [37,96] -> 133 [48,74] -> 122
3564: [36,99] -> 135 [44,81] -> 125 [54,66] -> 120
3570: [42,85] -> 127 [51,70] -> 121
3572: [38,94] -> 132 [47,76] -> 123
3588: [39,92] -> 131 [46,78] -> 124 [52,69] -> 121
3600: [40,90] -> 130 [45,80] -> 125 [48,75] -> 123 [50,72] -> 122 [60,60] -> 120
3608: [41,88] -> 129 [44,82] -> 126
3612: [42,86] -> 128 [43,84] -> 127
3626: [37,98] -> 135 [49,74] -> 123
3640: [40,91] -> 131 [52,70] -> 122 [56,65] -> 121
3648: [38,96] -> 134 [48,76] -> 124 [57,64] -> 121
3654: [42,87] -> 129 [58,63] -> 121
3666: [39,94] -> 133 [47,78] -> 125
3672: [51,72] -> 123 [54,68] -> 122
3680: [40,92] -> 132 [46,80] -> 126
3690: [41,90] -> 131 [45,82] -> 127
3696: [42,88] -> 130 [44,84] -> 128 [48,77] -> 125 [56,66] -> 122
3705: [39,95] -> 134 [57,65] -> 122
3720: [40,93] -> 133 [60,62] -> 122
3724: [38,98] -> 136 [49,76] -> 125
3726: [46,81] -> 127 [54,69] -> 123
3740: [44,85] -> 129 [55,68] -> 123
3744: [39,96] -> 135 [48,78] -> 126 [52,72] -> 124
3760: [40,94] -> 134 [47,80] -> 127
3762: [38,99] -> 137 [57,66] -> 123
3772: [41,92] -> 133 [46,82] -> 128
3780: [42,90] -> 132 [45,84] -> 129 [54,70] -> 124 [60,63] -> 123
3784: [43,88] -> 131 [44,86] -> 130
3800: [40,95] -> 135 [50,76] -> 126
3822: [39,98] -> 137 [42,91] -> 133 [49,78] -> 127
3825: [45,85] -> 130 [51,75] -> 126
3828: [44,87] -> 131 [58,66] -> 124
3840: [40,96] -> 136 [48,80] -> 128 [60,64] -> 124
3850: [50,77] -> 127 [55,70] -> 125
3854: [41,94] -> 135 [47,82] -> 129
3864: [42,92] -> 134 [46,84] -> 130 [56,69] -> 125
3870: [43,90] -> 133 [45,86] -> 131
3876: [51,76] -> 127 [57,68] -> 125
3888: [48,81] -> 129 [54,72] -> 126
3900: [50,78] -> 128 [52,75] -> 127 [60,65] -> 125
3906: [42,93] -> 135 [62,63] -> 125
3920: [40,98] -> 138 [49,80] -> 129 [56,70] -> 126
3936: [41,96] -> 137 [48,82] -> 130
3948: [42,94] -> 136 [47,84] -> 131
3956: [43,92] -> 135 [46,86] -> 132
3960: [40,99] -> 139 [44,90] -> 134 [45,88] -> 133 [55,72] -> 127 [60,66] -> 126
3969: [49,81] -> 130 [63,63] -> 126
3990: [42,95] -> 137 [57,70] -> 127
4002: [46,87] -> 133 [58,69] -> 127
4004: [44,91] -> 135 [52,77] -> 129
4018: [41,98] -> 139 [49,82] -> 131
4032: [42,96] -> 138 [48,84] -> 132 [56,72] -> 128 [63,64] -> 127
4042: [43,94] -> 137 [47,86] -> 133
4048: [44,92] -> 136 [46,88] -> 134
4050: [45,90] -> 135 [50,81] -> 131 [54,75] -> 129
4080: [48,85] -> 133 [51,80] -> 131 [60,68] -> 128
4092: [44,93] -> 137 [62,66] -> 128
4095: [45,91] -> 136 [63,65] -> 128
4104: [54,76] -> 130 [57,72] -> 129
4116: [42,98] -> 140 [49,84] -> 133
4128: [43,96] -> 139 [48,86] -> 134
4136: [44,94] -> 138 [47,88] -> 135
4140: [45,92] -> 137 [46,90] -> 136 [60,69] -> 129
4158: [42,99] -> 141 [54,77] -> 131 [63,66] -> 129
4160: [52,80] -> 132 [64,65] -> 129
4176: [48,87] -> 135 [58,72] -> 130
4180: [44,95] -> 139 [55,76] -> 131
4200: [50,84] -> 134 [56,75] -> 131 [60,70] -> 130
4212: [52,81] -> 133 [54,78] -> 132
4214: [43,98] -> 141 [49,86] -> 135
4224: [44,96] -> 140 [48,88] -> 136 [64,66] -> 130
4230: [45,94] -> 139 [47,90] -> 137
4275: [45,95] -> 140 [57,75] -> 132
4278: [46,93] -> 139 [62,69] -> 131
4284: [51,84] -> 135 [63,68] -> 131
4290: [55,78] -> 133 [65,66] -> 131
4312: [44,98] -> 142 [49,88] -> 137 [56,77] -> 133
4320: [45,96] -> 141 [48,90] -> 138 [54,80] -> 134 [60,72] -> 132
4324: [46,94] -> 140 [47,92] -> 139
4350: [50,87] -> 137 [58,75] -> 133
4356: [44,99] -> 143 [66,66] -> 132
4368: [48,91] -> 139 [52,84] -> 136 [56,78] -> 134
4400: [50,88] -> 138 [55,80] -> 135
4410: [45,98] -> 143 [49,90] -> 139 [63,70] -> 133
4416: [46,96] -> 142 [48,92] -> 140 [64,69] -> 133
4420: [52,85] -> 137 [65,68] -> 133
4455: [45,99] -> 144 [55,81] -> 136
4464: [48,93] -> 141 [62,72] -> 134
4480: [56,80] -> 136 [64,70] -> 134
4488: [51,88] -> 139 [66,68] -> 134
4500: [50,90] -> 140 [60,75] -> 135
4508: [46,98] -> 144 [49,92] -> 141
4512: [47,96] -> 143 [48,94] -> 142
4524: [52,87] -> 139 [58,78] -> 136
4536: [54,84] -> 138 [56,81] -> 137 [63,72] -> 135
4550: [50,91] -> 141 [65,70] -> 135
4554: [46,99] -> 145 [66,69] -> 135
4560: [48,95] -> 143 [57,80] -> 137 [60,76] -> 136
4590: [51,90] -> 141 [54,85] -> 139
4606: [47,98] -> 145 [49,94] -> 143
4608: [48,96] -> 144 [64,72] -> 136
4620: [55,84] -> 139 [60,77] -> 137 [66,70] -> 136
4650: [50,93] -> 143 [62,75] -> 137
4680: [52,90] -> 142 [60,78] -> 138 [65,72] -> 137
4692: [51,92] -> 143 [68,69] -> 137
4698: [54,87] -> 141 [58,81] -> 139
4704: [48,98] -> 146 [49,96] -> 145 [56,84] -> 140
4752: [48,99] -> 147 [54,88] -> 142 [66,72] -> 138
4760: [56,85] -> 141 [68,70] -> 138
4788: [57,84] -> 141 [63,76] -> 139
4800: [50,96] -> 146 [60,80] -> 140 [64,75] -> 139
4836: [52,93] -> 145 [62,78] -> 140
4845: [51,95] -> 146 [57,85] -> 142
4851: [49,99] -> 148 [63,77] -> 140
4860: [54,90] -> 144 [60,81] -> 141
4872: [56,87] -> 143 [58,84] -> 142
4896: [51,96] -> 147 [68,72] -> 140
4900: [50,98] -> 148 [70,70] -> 140
4914: [54,91] -> 145 [63,78] -> 141
4928: [56,88] -> 144 [64,77] -> 141
4940: [52,95] -> 147 [65,76] -> 141
4950: [50,99] -> 149 [55,90] -> 145 [66,75] -> 141
4968: [54,92] -> 146 [69,72] -> 141
4992: [52,96] -> 148 [64,78] -> 142
5005: [55,91] -> 146 [65,77] -> 142
5016: [57,88] -> 145 [66,76] -> 142
5022: [54,93] -> 147 [62,81] -> 143
5040: [56,90] -> 146 [60,84] -> 144 [63,80] -> 143 [70,72] -> 142
5096: [52,98] -> 150 [56,91] -> 147
5100: [60,85] -> 145 [68,75] -> 143
5130: [54,95] -> 149 [57,90] -> 147
5148: [52,99] -> 151 [66,78] -> 144
5184: [54,96] -> 150 [64,81] -> 145 [72,72] -> 144
5208: [56,93] -> 149 [62,84] -> 146
5220: [58,90] -> 148 [60,87] -> 147
5244: [57,92] -> 149 [69,76] -> 145
5280: [55,96] -> 151 [60,88] -> 148 [66,80] -> 146
5292: [54,98] -> 152 [63,84] -> 147
5320: [56,95] -> 151 [70,76] -> 146
5346: [54,99] -> 153 [66,81] -> 147
5376: [56,96] -> 152 [64,84] -> 148
5390: [55,98] -> 153 [70,77] -> 147
5394: [58,93] -> 151 [62,87] -> 149
5400: [60,90] -> 150 [72,75] -> 147
5440: [64,85] -> 149 [68,80] -> 148
5460: [60,91] -> 151 [65,84] -> 149 [70,78] -> 148
5472: [57,96] -> 153 [72,76] -> 148
5520: [60,92] -> 152 [69,80] -> 149
5544: [56,99] -> 155 [63,88] -> 151 [66,84] -> 150 [72,77] -> 149
5568: [58,96] -> 154 [64,87] -> 151
5580: [60,93] -> 153 [62,90] -> 152
5670: [63,90] -> 153 [70,81] -> 151
5700: [60,95] -> 155 [75,76] -> 151
5742: [58,99] -> 157 [66,87] -> 153
5760: [60,96] -> 156 [64,90] -> 154 [72,80] -> 152
5796: [63,92] -> 155 [69,84] -> 153
5850: [65,90] -> 155 [75,78] -> 153
5880: [60,98] -> 158 [70,84] -> 154
5940: [60,99] -> 159 [66,90] -> 156
5952: [62,96] -> 158 [64,93] -> 157
6006: [66,91] -> 157 [77,78] -> 155
6048: [63,96] -> 159 [72,84] -> 156
6072: [66,92] -> 158 [69,88] -> 157
6080: [64,95] -> 159 [76,80] -> 156
6120: [68,90] -> 158 [72,85] -> 157
6138: [62,99] -> 161 [66,93] -> 159
6160: [70,88] -> 158 [77,80] -> 157
6237: [63,99] -> 162 [77,81] -> 158
6240: [65,96] -> 161 [78,80] -> 158
6300: [70,90] -> 160 [75,84] -> 159
6336: [64,99] -> 163 [66,96] -> 162 [72,88] -> 160
6370: [65,98] -> 163 [70,91] -> 161
6460: [68,95] -> 163 [76,85] -> 161
6468: [66,98] -> 164 [77,84] -> 161
6480: [72,90] -> 162 [80,81] -> 161
6552: [72,91] -> 163 [78,84] -> 162
6624: [69,96] -> 165 [72,92] -> 164
6720: [70,96] -> 166 [80,84] -> 164
6840: [72,95] -> 167 [76,90] -> 166
6930: [70,99] -> 169 [77,90] -> 167
7056: [72,98] -> 170 [84,84] -> 168
7128: [72,99] -> 171 [81,88] -> 169
7200: [75,96] -> 171 [80,90] -> 170
7392: [77,96] -> 173 [84,88] -> 172
7644: [78,98] -> 176 [84,91] -> 175
7920: [80,99] -> 179 [88,90] -> 178
Negativ-Beispiel: Wäre Gauß als Produkt die Zahl 10 genannt worden, hätte er direkt die beiden Faktoren 2 und 5 nennen können.
2) Euler: „Ja, das war mir klar.“
Euler wurde demnach eine Summe genannt, die sich außschließlich aus solchen Summandenpaaren zusammensetzt, deren sämtliche Produkte sich durch mehr als eine Weise bilden lassen, da er sonst nicht sicher hätte voraussagen können, dass Gauß die Zahlen nicht finden kann. Folgende Summen erfüllen diese Bedingung:
11: [2,9] -> 18 [3,8] -> 24 [4,7] -> 28 [5,6] -> 30
17: [2,15] -> 30 [3,14] -> 42 [4,13] -> 52 [5,12] -> 60 [6,11] -> 66 [7,10] -> 70 [8,9] -> 72
23: [2,21] -> 42 [3,20] -> 60 [4,19] -> 76 [5,18] -> 90 [6,17] -> 102 [7,16] -> 112 [8,15] -> 120 [9,14] -> 126 [10,13] -> 130 [11,12] -> 132
27: [2,25] -> 50 [3,24] -> 72 [4,23] -> 92 [5,22] -> 110 [6,21] -> 126 [7,20] -> 140 [8,19] -> 152 [9,18] -> 162 [10,17] -> 170 [11,16] -> 176 [12,15] -> 180 [13,14] -> 182
29: [2,27] -> 54 [3,26] -> 78 [4,25] -> 100 [5,24] -> 120 [6,23] -> 138 [7,22] -> 154 [8,21] -> 168 [9,20] -> 180 [10,19] -> 190 [11,18] -> 198 [12,17] -> 204 [13,16] -> 208 [14,15] -> 210
35: [2,33] -> 66 [3,32] -> 96 [4,31] -> 124 [5,30] -> 150 [6,29] -> 174 [7,28] -> 196 [8,27] -> 216 [9,26] -> 234 [10,25] -> 250 [11,24] -> 264 [12,23] -> 276 [13,22] -> 286 [14,21] -> 294 [15,20] -> 300 [16,19] -> 304 [17,18] -> 306
37: [2,35] -> 70 [3,34] -> 102 [4,33] -> 132 [5,32] -> 160 [6,31] -> 186 [7,30] -> 210 [8,29] -> 232 [9,28] -> 252 [10,27] -> 270 [11,26] -> 286 [12,25] -> 300 [13,24] -> 312 [14,23] -> 322 [15,22] -> 330 [16,21] -> 336 [17,20] -> 340 [18,19] -> 342
41: [2,39] -> 78 [3,38] -> 114 [4,37] -> 148 [5,36] -> 180 [6,35] -> 210 [7,34] -> 238 [8,33] -> 264 [9,32] -> 288 [10,31] -> 310 [11,30] -> 330 [12,29] -> 348 [13,28] -> 364 [14,27] -> 378 [15,26] -> 390 [16,25] -> 400 [17,24] -> 408 [18,23] -> 414 [19,22] -> 418 [20,21] -> 420
47: [2,45] -> 90 [3,44] -> 132 [4,43] -> 172 [5,42] -> 210 [6,41] -> 246 [7,40] -> 280 [8,39] -> 312 [9,38] -> 342 [10,37] -> 370 [11,36] -> 396 [12,35] -> 420 [13,34] -> 442 [14,33] -> 462 [15,32] -> 480 [16,31] -> 496 [17,30] -> 510 [18,29] -> 522 [19,28] -> 532 [20,27] -> 540 [21,26] -> 546 [22,25] -> 550 [23,24] -> 552
53: [2,51] -> 102 [3,50] -> 150 [4,49] -> 196 [5,48] -> 240 [6,47] -> 282 [7,46] -> 322 [8,45] -> 360 [9,44] -> 396 [10,43] -> 430 [11,42] -> 462 [12,41] -> 492 [13,40] -> 520 [14,39] -> 546 [15,38] -> 570 [16,37] -> 592 [17,36] -> 612 [18,35] -> 630 [19,34] -> 646 [20,33] -> 660 [21,32] -> 672 [22,31] -> 682 [23,30] -> 690 [24,29] -> 696 [25,28] -> 700 [26,27] -> 702
Negativ-Beispiel: Wäre Euler als Summe die Zahl 10 genannt worden, dann hätte Gauß beispielsweise die Zahl 21 (als das Produkt aus 3 mal 7) genannt bekommen können, und dieser hätte direkt wegen der Eindeutigkeit der beiden Faktoren dieselben sicher nennen können. Euler hätte somit nicht voraussagen können, dass Gauß die Zahlen nicht finden kann.
3) Gauß: „Ah, jetzt kenne ich die beiden Zahlen.“
Jetzt werden die durch die erste Bedingung ermittelten roten Summen interessant, die sich ja aus den jeweiligen möglichen Faktoren der Produkte dort ergeben haben. Pro Produkt haben wir eine Menge von Summen erhalten, und in jeweils einer solchen Menge muss sich nun genau eine der Summen befinden, die durch die zweite Bedingung gefunden worden sind, denn nur dann kann sich Gauß des Zahlenpaares auch wirklich sicher sein. Folgende Produkte bleiben dann noch übrig:
18: [2,9] -> 11
24: [3,8] -> 11
28: [4,7] -> 11
50: [2,25] -> 27
52: [4,13] -> 17
54: [2,27] -> 29
76: [4,19] -> 23
92: [4,23] -> 27
96: [3,32] -> 35
100: [4,25] -> 29
110: [5,22] -> 27
112: [7,16] -> 23
114: [3,38] -> 41
124: [4,31] -> 35
130: [10,13] -> 23
138: [6,23] -> 29
140: [7,20] -> 27
148: [4,37] -> 41
152: [8,19] -> 27
154: [7,22] -> 29
160: [5,32] -> 37
162: [9,18] -> 27
168: [8,21] -> 29
170: [10,17] -> 27
172: [4,43] -> 47
174: [6,29] -> 35
176: [11,16] -> 27
182: [13,14] -> 27
186: [6,31] -> 37
190: [10,19] -> 29
198: [11,18] -> 29
204: [12,17] -> 29
208: [13,16] -> 29
216: [8,27] -> 35
232: [8,29] -> 37
234: [9,26] -> 35
238: [7,34] -> 41
240: [5,48] -> 53
246: [6,41] -> 47
250: [10,25] -> 35
252: [9,28] -> 37
270: [10,27] -> 37
276: [12,23] -> 35
280: [7,40] -> 47
282: [6,47] -> 53
288: [9,32] -> 41
294: [14,21] -> 35
304: [16,19] -> 35
306: [17,18] -> 35
310: [10,31] -> 41
336: [16,21] -> 37
340: [17,20] -> 37
348: [12,29] -> 41
360: [8,45] -> 53
364: [13,28] -> 41
370: [10,37] -> 47
378: [14,27] -> 41
390: [15,26] -> 41
400: [16,25] -> 41
408: [17,24] -> 41
414: [18,23] -> 41
418: [19,22] -> 41
430: [10,43] -> 53
442: [13,34] -> 47
480: [15,32] -> 47
492: [12,41] -> 53
496: [16,31] -> 47
510: [17,30] -> 47
520: [13,40] -> 53
522: [18,29] -> 47
532: [19,28] -> 47
540: [20,27] -> 47
550: [22,25] -> 47
552: [23,24] -> 47
570: [15,38] -> 53
592: [16,37] -> 53
612: [17,36] -> 53
630: [18,35] -> 53
646: [19,34] -> 53
660: [20,33] -> 53
672: [21,32] -> 53
682: [22,31] -> 53
690: [23,30] -> 53
696: [24,29] -> 53
700: [25,28] -> 53
702: [26,27] -> 53
Negativ-Beispiel: Hätte man Gauß die Zahl 30 und Euler die Zahl 17 genannt, dann hätte Gauß die beiden Faktoren nicht direkt bestimmen können (erste Bedingung erfüllt), und Euler hätte das auch voraussehen können (zweite Bedingung erfüllt). Gauß hätte zwar anschließend das Zahlenpaar [3,10] ausschließen können, aber weiter nicht entscheiden können, welche Summe der verbleibenden Zahlenpaare [2,15] und [5,6] Euler genannt bekommen hätte, und ihm wäre eine genaue Bestimmung der beiden Zahlen somit verwehrt geblieben.
4) Euler: „Also, dann kenne ich sie jetzt auch.“
Folglich darf unter den durch die zweite Bedingung ermittelten Summen nur diejenige übrigbleiben, die sich nur durch genau ein Produkt aus den Zahlenpaaren ergibt, die durch die dritte Bedingung übrig geblieben sind, denn sonst könnte Euler das Zahlenpaar nicht eindeutig benennen.
11: Kann auf 3 verschiedene Weisen erzeugt werden.
17: Kann auf 1 verschiedene Weise erzeugt werden.
23: Kann auf 3 verschiedene Weisen erzeugt werden.
27: Kann auf 9 verschiedene Weisen erzeugt werden.
29: Kann auf 9 verschiedene Weisen erzeugt werden.
35: Kann auf 10 verschiedene Weisen erzeugt werden.
37: Kann auf 7 verschiedene Weisen erzeugt werden.
41: Kann auf 13 verschiedene Weisen erzeugt werden.
47: Kann auf 13 verschiedene Weisen erzeugt werden.
53: Kann auf 18 verschiedene Weisen erzeugt werden.
Negativ-Beispiel: Wäre Euler als Summe die Zahl 11 genannt worden, dann hätte er aufgrund der vorhergehenden Einschränkungen drei verschiedene Zahlenpaare ([2,9], [3,8] und [4,7]) zur Auswahl gehabt, deren jeweiliges Produkt Gauß genannt worden sein könnte, und er selbst hätte die beiden Zahlen nicht sicher nennen können.
Lösung:
Das Zahlenpaar lautet:
[4,13]
Gegenprobe:
Luzifer nannte demnach Gauß die Zahl 52 und Euler die Zahl 17. Nun sollen die Gedankengänge der beiden Mathematiker in ihrem Dialog Schritt für Schritt nachgezeichnet werden.
Schritt 1:
Gauß: „Ich kann die zwei Zahlen nicht finden!“
Produkt = 52: [2,26] -> 28 [4,13] -> 17
Gauß kann also die zwei Zahlen tatsächlich nicht finden, denn es gibt mehr als ein Faktorenpaar, mit dem sich das Produkt bilden lässt.
Schritt 2:
Euler: „Ja, das war mir klar.“
Summe = 17:
[2,15] -> 30 = 3 x 10 (anders faktorisierbar)
[3,14] -> 42 = 2 x 21 (anders faktorisierbar)
[4,13] -> 52 = 2 x 26 (anders faktorisierbar)
[5,12] -> 60 = 2 x 30 (anders faktorisierbar)
[6,11] -> 66 = 2 x 33 (anders faktorisierbar)
[7,10] -> 70 = 2 x 35 (anders faktorisierbar)
[8,9] -> 72 = 2 x 36 (anders faktorisierbar)
Euler kann also tatsächlich wissen, dass Gauß die zwei Zahlen nicht kennt, denn alle Produkte, die sich aus Eulers genannter Zahl herleiten lassen, können auch auf andere Weise faktorisiert werden.
Schritt 3:
Gauß: „Ah, jetzt kenne ich die beiden Zahlen.“
Überprüfung der Summe = 28: [2,26] -> 52 = 4 x 13 (anders faktorisierbar)[3,25] -> 75 = 5 x 15 (anders faktorisierbar)[4,24] -> 96 = 2 x 48 (anders faktorisierbar)[5,23] -> 115 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)Überprüfung der Summe = 17: [2,15] -> 30 = 3 x 10 (anders faktorisierbar)[3,14] -> 42 = 2 x 21 (anders faktorisierbar)[4,13] -> 52 = 2 x 26 (anders faktorisierbar)[5,12] -> 60 = 2 x 30 (anders faktorisierbar)[6,11] -> 66 = 2 x 33 (anders faktorisierbar)[7,10] -> 70 = 2 x 35 (anders faktorisierbar)[8,9] -> 72 = 2 x 36 (anders faktorisierbar)
Gauß kennt nun tatächlich die beiden Zahlen, denn Euler konnte nur die Summe 17 genannt worden sein. Die Produkte, die sich aus der anderen Summe herleiten lassen, sind nicht alle auch anders faktorisierbar, was für Eulers Wissen aber notwendig gewesen wäre.
Schritt 4:
Euler: „Also, dann kenne ich sie jetzt auch.“
Überprüfung des Produkts = 30: [2,15] -> Summe = 17
[2,15] -> 30 = 3 x 10 (anders faktorisierbar)
[3,14] -> 42 = 2 x 21 (anders faktorisierbar)
[4,13] -> 52 = 2 x 26 (anders faktorisierbar)
[5,12] -> 60 = 2 x 30 (anders faktorisierbar)
[6,11] -> 66 = 2 x 33 (anders faktorisierbar)
[7,10] -> 70 = 2 x 35 (anders faktorisierbar)
[8,9] -> 72 = 2 x 36 (anders faktorisierbar)
-> Möglichkeit für Gauß [3,10] -> Summe = 13
[2,11] -> 22 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [5,6] -> Summe = 11
[2,9] -> 18 = 3 x 6 (anders faktorisierbar)
[3,8] -> 24 = 2 x 12 (anders faktorisierbar)
[4,7] -> 28 = 2 x 14 (anders faktorisierbar)
[5,6] -> 30 = 2 x 15 (anders faktorisierbar)
-> Möglichkeit für Gauß --> insgesamt 2 Möglichkeiten für Gauß, folglich für ihn nicht entscheidbar... Überprüfung des Produkts = 42: [2,21] -> Summe = 23
[2,21] -> 42 = 3 x 14 (anders faktorisierbar)
[3,20] -> 60 = 2 x 30 (anders faktorisierbar)
[4,19] -> 76 = 2 x 38 (anders faktorisierbar)
[5,18] -> 90 = 2 x 45 (anders faktorisierbar)
[6,17] -> 102 = 2 x 51 (anders faktorisierbar)
[7,16] -> 112 = 2 x 56 (anders faktorisierbar)
[8,15] -> 120 = 2 x 60 (anders faktorisierbar)
[9,14] -> 126 = 2 x 63 (anders faktorisierbar)
[10,13] -> 130 = 2 x 65 (anders faktorisierbar)
[11,12] -> 132 = 2 x 66 (anders faktorisierbar)
-> Möglichkeit für Gauß [3,14] -> Summe = 17 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß [6,7] -> Summe = 13 (hat Euler oben schon berechnet)-> keine Möglichkeit für Gauß --> insgesamt 2 Möglichkeiten für Gauß, folglich für ihn nicht entscheidbar... Überprüfung des Produkts = 52: [2,26] -> Summe = 28
[2,26] -> 52 = 4 x 13 (anders faktorisierbar)
[3,25] -> 75 = 5 x 15 (anders faktorisierbar)
[4,24] -> 96 = 2 x 48 (anders faktorisierbar)
[5,23] -> 115 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [4,13] -> Summe = 17 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß --> genau 1 Möglichkeit für Gauß: Folglich hätte er die Zahlen ermitteln können!Überprüfung des Produkts = 60: [2,30] -> Summe = 32
[2,30] -> 60 = 3 x 20 (anders faktorisierbar)
[3,29] -> 87 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [3,20] -> Summe = 23 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß [4,15] -> Summe = 19
[2,17] -> 34 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [5,12] -> Summe = 17 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß [6,10] -> Summe = 16
[2,14] -> 28 = 4 x 7 (anders faktorisierbar)
[3,13] -> 39 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß --> insgesamt 2 Möglichkeiten für Gauß, folglich für ihn nicht entscheidbar... Überprüfung des Produkts = 66: [2,33] -> Summe = 35
[2,33] -> 66 = 3 x 22 (anders faktorisierbar)
[3,32] -> 96 = 2 x 48 (anders faktorisierbar)
[4,31] -> 124 = 2 x 62 (anders faktorisierbar)
[5,30] -> 150 = 2 x 75 (anders faktorisierbar)
[6,29] -> 174 = 2 x 87 (anders faktorisierbar)
[7,28] -> 196 = 2 x 98 (anders faktorisierbar)
[8,27] -> 216 = 3 x 72 (anders faktorisierbar)
[9,26] -> 234 = 3 x 78 (anders faktorisierbar)
[10,25] -> 250 = 5 x 50 (anders faktorisierbar)
[11,24] -> 264 = 3 x 88 (anders faktorisierbar)
[12,23] -> 276 = 3 x 92 (anders faktorisierbar)
[13,22] -> 286 = 11 x 26 (anders faktorisierbar)
[14,21] -> 294 = 3 x 98 (anders faktorisierbar)
[15,20] -> 300 = 4 x 75 (anders faktorisierbar)
[16,19] -> 304 = 4 x 76 (anders faktorisierbar)
[17,18] -> 306 = 6 x 51 (anders faktorisierbar)
-> Möglichkeit für Gauß [3,22] -> Summe = 25
[2,23] -> 46 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [6,11] -> Summe = 17 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß --> insgesamt 2 Möglichkeiten für Gauß, folglich für ihn nicht entscheidbar... Überprüfung des Produkts = 70: [2,35] -> Summe = 37
[2,35] -> 70 = 5 x 14 (anders faktorisierbar)
[3,34] -> 102 = 2 x 51 (anders faktorisierbar)
[4,33] -> 132 = 2 x 66 (anders faktorisierbar)
[5,32] -> 160 = 2 x 80 (anders faktorisierbar)
[6,31] -> 186 = 2 x 93 (anders faktorisierbar)
[7,30] -> 210 = 3 x 70 (anders faktorisierbar)
[8,29] -> 232 = 4 x 58 (anders faktorisierbar)
[9,28] -> 252 = 3 x 84 (anders faktorisierbar)
[10,27] -> 270 = 3 x 90 (anders faktorisierbar)
[11,26] -> 286 = 13 x 22 (anders faktorisierbar)
[12,25] -> 300 = 4 x 75 (anders faktorisierbar)
[13,24] -> 312 = 4 x 78 (anders faktorisierbar)
[14,23] -> 322 = 7 x 46 (anders faktorisierbar)
[15,22] -> 330 = 5 x 66 (anders faktorisierbar)
[16,21] -> 336 = 4 x 84 (anders faktorisierbar)
[17,20] -> 340 = 4 x 85 (anders faktorisierbar)
[18,19] -> 342 = 6 x 57 (anders faktorisierbar)
-> Möglichkeit für Gauß [5,14] -> Summe = 19 (hat Euler oben schon berechnet)-> keine Möglichkeit für Gauß [7,10] -> Summe = 17 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß --> insgesamt 2 Möglichkeiten für Gauß, folglich für ihn nicht entscheidbar... Überprüfung des Produkts = 72: [2,36] -> Summe = 38
[2,36] -> 72 = 3 x 24 (anders faktorisierbar)
[3,35] -> 105 = 5 x 21 (anders faktorisierbar)
[4,34] -> 136 = 2 x 68 (anders faktorisierbar)
[5,33] -> 165 = 3 x 55 (anders faktorisierbar)
[6,32] -> 192 = 2 x 96 (anders faktorisierbar)
[7,31] -> 217 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [3,24] -> Summe = 27
[2,25] -> 50 = 5 x 10 (anders faktorisierbar)
[3,24] -> 72 = 2 x 36 (anders faktorisierbar)
[4,23] -> 92 = 2 x 46 (anders faktorisierbar)
[5,22] -> 110 = 2 x 55 (anders faktorisierbar)
[6,21] -> 126 = 2 x 63 (anders faktorisierbar)
[7,20] -> 140 = 2 x 70 (anders faktorisierbar)
[8,19] -> 152 = 2 x 76 (anders faktorisierbar)
[9,18] -> 162 = 2 x 81 (anders faktorisierbar)
[10,17] -> 170 = 2 x 85 (anders faktorisierbar)
[11,16] -> 176 = 2 x 88 (anders faktorisierbar)
[12,15] -> 180 = 2 x 90 (anders faktorisierbar)
[13,14] -> 182 = 2 x 91 (anders faktorisierbar)
-> Möglichkeit für Gauß [4,18] -> Summe = 22
[2,20] -> 40 = 4 x 10 (anders faktorisierbar)
[3,19] -> 57 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [6,12] -> Summe = 18
[2,16] -> 32 = 4 x 8 (anders faktorisierbar)
[3,15] -> 45 = 5 x 9 (anders faktorisierbar)
[4,14] -> 56 = 2 x 28 (anders faktorisierbar)
[5,13] -> 65 (nicht anders faktorisierbar -> Summenzahl muss nicht weiter überprüft werden!)
-> keine Möglichkeit für Gauß [8,9] -> Summe = 17 (hat Euler oben schon berechnet)-> Möglichkeit für Gauß --> insgesamt 2 Möglichkeiten für Gauß, folglich für ihn nicht entscheidbar...
Euler kennt nun tatsächlich auch die beiden Zahlen, denn Gauß konnte nur zum Produkt 52 die Zahlen eindeutig ermitteln.
Bestätigung:
Die Zahlen 4 und 13 erfüllen somit die im Dialog implizierten Bedingungen. (Ob es auch andere Zahlenpaare gibt, die diese Bedingungen erfüllen, ist damit aber noch nicht gezeigt.)
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