# Knödelovo število

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Knödelovo število je v teoriji števil za dano naravno število n sestavljeno število m z lastnostjo, da za vsak i < m, ki je m tuj, velja:

$i^{m - n} \equiv 1 \mod m \; .$

Množica takšnih celih števil za n se potem imenuje množica Knödelovih števil Kn.

Razpredelnica podaja prve elemente množic Kn za 0 < n < 26.[1]

n Kn OEIS
1 561, 1105, 1729, 2465, 2821, 6601, 8911 A002997
2 4, 6, 8, 10, 12, 14, 22, 24, 26, 30, 34, 38, 46, 56, 58, 62, 74, 82, 86, 94 A050990
3 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93 A033553
4 6, 8, 12, 16, 20, 24, 28, 40, 44, 48, 52, 60, 68, 76, 80, 92 A050992
5 25, 65, 85, 145, 165, 185, 205, 265, 305, 365, 445, 485, 505, 545, 565, 685, 745, 785, 825, 865, 905, 965, 985 A050993
6 8, 10, 12, 18, 24, 30, 36, 42, 66, 72, 78, 84, 90
7 9, 15, 49, 91, 133, 217, 259, 301, 427, 469, 511, 553, 679, 721, 763, 889, 973
8 10, 12, 14, 16, 20, 24, 32, 40, 48, 56, 60, 80, 88, 96
9 21, 27, 45, 63, 99, 105, 117, 153, 171, 189, 207, 261, 273, 279, 333, 369, 387, 423, 429, 477, 513, 531, 549, 585, 603, 639, 657, 711, 747, 801, 873, 909, 927, 945, 963, 981
10 12, 24, 28, 30, 50, 70, 110, 130, 150, 170, 190, 230, 290, 310, 330, 370, 410, 430, 442, 470, 530, 532, 550, 590, 610, 670, 710, 730, 790, 830, 890, 910, 970
11 15, 35, 121, 341, 451, 455, 671, 781
12 14, 16, 18, 20, 22, 24, 36, 40, 42, 48, 60, 72, 80, 84
13 14, 15, 33, 169, 481, 793, 805, 949
14 15, 16, 18, 24, 26, 30, 44, 56, 98, 182, 264, 266, 392, 434, 510, 518, 602, 854, 938
15 16, 21, 39, 55, 63, 75, 195, 255, 275, 315, 435, 495, 555, 615, 795, 819, 915, 975
16 18, 20, 24, 28, 32, 40, 48, 52, 60, 64, 66, 80, 96
17 65, 77, 289, 665, 1649, 1921
18 20, 24, 30, 34, 36, 42, 54, 72, 78, 84, 88, 90
19 21, 51, 91, 361, 595, 703, 1387, 1955
20 22, 24, 38, 40, 48, 56, 60, 68, 80, 100
21 45, 57, 63, 85, 105, 117, 147, 231, 273, 357, 399, 441, 483, 585, 609, 651, 741, 777, 861, 903, 987
22 24, 28, 30, 70, 76, 102, 130, 132, 242, 682, 902, 910
23 25, 33, 35, 95, 119, 143, 455, 529
24 26, 30, 32, 36, 40, 42, 44, 46, 48, 60, 72, 80, 84, 96
25 27, 69, 125, 133, 165, 325, 385, 425, 725, 825, 925

A. Makowski je v letih 1962/63 dokazal, da obstaja neskončno mnogo elementov množic Kn za $n \ge 2$.

## Viri

• Makowski, A. (1963). "Generalization of Morrow's D-Numbers". Simon Stevin 36: 71.