Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$

Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation.

Is there a largest $n$ with the desired property for any $m$? If so, define $n(m)$ to be this number.

Examples :

$$m=2 \rightarrow 2^{168} = 374144419156711147060143317175368453031918731001856$$

does not contain the digit $2$.

All the powers above up to $2^{10000}$ conatin all the digits, so $2^{168}$ seems to be the biggest power with the desired property.

$$m=3 \rightarrow 3^{106} = 375710212613636260325580163599137907799836383538729$$

does not contain the digit $4$.

All the powers above up to $3^{10000}$ contain all the digits, so $3^{106}$ seems to be the biggest power with the desired property.

So, probably $n(2)=168$ and $n(3)=106$ hold.

Is $n(m)$ defined for any $m$, and if yes, can reasonably sharp bounds be given?

Solution 1:

Some obesrvations:

I have tested $2^n$ up to $n$=250000 and found that $n(2)=168$ still holds. Having played with this problem a lot I conjecture that $n(m)\le168$ for all $m$.

I went a little bit further down the road and found the following values for numbers up to 340. All values have been obtained by calcuating all powers up to power 50000 and getting the biggest power with a digit missing in the result.

I have also checked values $n(m)$ in the range $m=10^6...10^9$ and found local maximums to go down from around 20 to 10. And from time to time you can find a real gem:

$32886598^{26} = \\ 2770082411 \ 7701047414 \ 3812147939 \ 0119763476 \ 3327029432 \\ 3084767371 \ 7070016012 \ 4780829912 \ 9634078101 \ 6224929090 \\ 6339478284 \ 9104901979 \ 0146722638 \ 7896000449 \ 7946341749 \\ 3932606670 \ 3148399037 \ 2721668024 \ 4923962610 \ 417664$

196 digits with digit 5 missing.

(2,168), (3,106), (4,84), (5,65), (6,64), (7,61), (8,56), (9,53), 
(11,41), (12,51), (13,37), (14,34), (15,34), (16,42), (17,27), (18,25), (19,44), 
(21,29), (22,24), (23,50), (24,23), (25,29), (26,31), (27,), (8), (28,28), (29,45), 
(31,28), (32,18), (33,24), (34,34), (35,18), (36,32), (37,25), (38,17), (39,41), 
(41,23), (42,19), (43,20), (44,29), (45,39), (46,32), (47,15), (48,29), (49,16), 
(51,29), (52,29), (53,30), (54,18), (55,17), (56,33), (57,19), (58,31), (59,27), 
(61,26), (62,19), (63,24), (64,28), (65,17), (66,15), (67,21), (68,25), (69,13), 
(71,25), (72,39), (73,17), (74,19), (75,19), (76,21), (77,24), (78,19), (79,30), 
(81,26), (82,25), (83,19), (84,27), (85,17), (86,25), (87,23), (88,23), (89,32), 
(91,23), (92,22), (93,16), (94,18), (95,26), (96,20), (97,24), (98,20), (99,21), 
(101,18), (102,17), (103,42), (104,28), (105,29), (106,21), (107,22), (108,17), (109,31), 
(111,32), (112,23), (113,19), (114,16), (115,30), (116,16), (117,17), (118,20), (119,26), 
(121,19), (122,23), (123,16), (124,13), (125,18), (126,17), (127,23), (128,24), (129,16), 
(131,16), (132,22), (133,18), (134,21), (135,16), (136,34), (137,27), (138,12), (135,16), (136,34), (137,27), (138,12), (139,14), 
(141,20), (142,19), (143,18), (144,20), (145,12), (146,17), (147,15), (148,16), (149,11), 
(151,14), (152,9), (153,10), (154,14), (155,18), (156,21), (157,20), (158,19), (159,30), 
(161,13), (162,19), (163,16), (164,26), (165,15), (166), (18), (167,13), (168,12), (169,15), 
(171,17), (172,19), (173,17), (174,12), (175,23), (176,21), (177,15), (178,13), (179,17), 
(181,11), (182,9), (183,10), (184,14), (185,24), (186,39), (187,17), (188,9), (189,12), 
(191,21), (192,12), (193,13), (194,20), (195,12), (196,17), (197,34), (198,20), (199,16), 
(201,12), (202,13), (203,13), (204,17), (205,22), (206,15), (207,21), (208,19), (209,16), 
(211,14), (212,22), (213,17), (214,17), (215,31), (216,15), (217,12), (218,17), (219,13), 
(221,16), (222,14), (223,18), (224,16), (225,17), (226,13), (227,27), (228,13), (229,18), 
(231,20), (232,15), (233,21), (234), (15), (235,20), (236,15), (237,25), (238,18), (239,16), 
(241,13), (242,26), (243,20), (244,27), (245,12), (246,25), (247,15), (248,10), (249,14), 
(251,11), (252,14), (253,11), (254,14), (255,28), (256,), (1), (257,20), (258,16), (259,24), 
(261,17), (262,19), (263,20), (264,15), (265,11), (266,20), (267,17), (268,14), (269,12), 
(271,36), (272,15), (273,18), (274,14), (275,13), (276,9), (277,17), (278,11), (279,13), 
(281,14), (282,14), (283,21), (284,13), (285,27), (286,13), (287,13), (288,18), (289,13), 
(291,18), (292,13), (293,14), (294,18), (295,8), (296,23), (297,25), (298,15), (299,15), 
(301,22), (3), (2,17), (303,19), (304,13), (305,10), (306,8), (307,11), (308,20), (309,12), 
(311,16), (312,15), (313,14), (314,17), (315,19), (316,14), (317,20), (318,12), (319,12), 
(321,9), (322,13), (323,11), (324,11), (325,9), (326,22), (327,13), (328,23), (329,15), 
(331,28), (332,18), (333,16), (334,13) ...

If you are interested in the length of the result, a few items from this list will "raise the bar":

$2^{168}$ has 51 digits.

$7^{61}$ has 52 digits.

$12^{51}$ has 56 digits.

$19^{44}$ has 57 digits.

$23^{50}$ has 69 digits.

$72^{39}$ has 73 digits.

$103^{42}$ has 85 digits.

$186^{39}$ has 89 digits.

$349^{39}$ has 100 digits.

$476^{41}$ has 110 digits.

$1955^{39}$ has 129 digits.

$42806^{30}$ has 139 digits.

$165541^{27}$ has 141 digits.

$191131^{27}$ has 143 digits.

$700419^{25}$ has 147 digits

$700419^25$ has 147 digits.

$874395^{27}$ has 161 digits.

$4408232^{25}$ has 167 digits.

$5397917^{27}$ has 182 digits.

$8751594^{27}$ has 188 digits.

$32886598^{26}$ has 196 digits

$54013149^{28}$ has 217 digits

$1274902129^{24}$ has 219 digits

$1337169719^{24}$ has 220 digits

EDIT: Current record:

$1419213312^{25}$ has 229 digits and no digit 7