Knödlovo število

Knödlovo 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:

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

Množica takšnih celih števil za n se potem imenuje množica Knödlovih števil K_n. Števila se imenujejo po avstrijskem matematiku in računalnikarju Walterju Knödlu.

K₁ so Carmichaelova števila.

Razpredelnica podaja prve elemente množic K_n 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 K_n za ${\displaystyle n\geq 2}$.

Sklici

1. Table of small Knödel numbers

Viri

• Makowski, A. (1963), »Generalization of Morrow's D-Numbers«, Simon Stevin, 36: 71
• Ribenboim, Paulo (1989), The New Book of Prime Number Records, New York: Springer-Verlag, str. 101, ISBN 978-0-387-94457-9

Zunanje povezave

• Weisstein, Eric Wolfgang. »Knoedel Numbers«. MathWorld.