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Comparing Different Numeral Bases Under Different Mathematical Conditions

Jamie Stroud

Table of contents

Introduction

The purpose of this article is to compare different numeral bases under different mathematical conditions to see if any patterns or interesting phenomena arise under particular numeral bases.


Quadratics

Smallest numbers that are perfect squares
Oct Dec Duo
4 4 4
20 16 14
100 64 54
400 256 194
2000 1024 714
1 0000 4096 2454
4 0000 1 6384 9594
20 0000 6 5536 3 1Ɛ14
100 0000 26 2144 10 7854
400 0000 104 8576 42 6994
2000 0000 419 4304 14ᘔ 3314
1 0000 0000 1677 7216 575 1054

Smallest numbers that are perfect cubes
Oct Dec Duo
10 8 8
100 64 54
1000 512 368
1 0000 4096 2454
10 0000 3 2768 1 6Ɛ68
100 0000 26 2144 10 7854
1000 0000 209 7152 85 1768
1 0000 0000 1677 7216 575 1054

Smallest numbers that are both perfect squares and perfect cubes
Oct Dec Duo
100 64 54
1 0000 4096 2454
100 0000 26 2144 10 7854
1 0000 0000 1677 7216 575 1054

Findings


Pi

Oct Dec Duo
3.1103755242102643021514230630505600670163211220111602105147630720 3.1415926535897932384626433832795028841971693993751058209749445923 3.184809493Ɛ918664573ᘔ6211ƐƐ151551ᘔ05729290ᘔ7809ᘔ492742140ᘔ60ᘔ5525

Findings


Primes

Prime numbers
Oct Dec Duo
2 2 2
3 3 3
5 5 5
7 7 7
13 11 Ɛ
15 13 11
21 17 15
23 19 17
27 23
35 29 25
37 31 27
45 37 31
51 41 35
53 43 37
57 47
65 53 45
73 59
75 61 51
103 67 57
107 71
111 73 61
117 79 67
123 83
131 89 75
141 97 81
145 101 85
147 103 87
153 107
155 109 91
161 113 95
177 127 ᘔ7
203 131 ᘔƐ
211 137 Ɛ5
213 139 Ɛ7
225 149 105
227 151 107
235 157 111
243 163 117
247 167 11Ɛ
255 173 125
263 179 12Ɛ
265 181 131
277 191 13Ɛ
301 193 141
305 197 145
307 199 147
323 211 157
337 223 167
343 227 16Ɛ
345 229 171
351 233 175
357 239 17Ɛ
361 241 181
373 251 18Ɛ
401 257 195
407 263 19Ɛ
415 269 1ᘔ5
417 271 1ᘔ7
425 277 1Ɛ1
431 281 1Ɛ5
433 283 1Ɛ7
445 293 205
463 307 217
467 311 21Ɛ
471 313 221
475 317 225
513 331 237
521 337 241
533 347 24Ɛ
535 349 251
541 353 255
547 359 25Ɛ
557 367 267
565 373 271
573 379 277
577 383 27Ɛ
605 389 285
615 397 291
621 401 295
631 409 2ᘔ1
643 419 2ᘔƐ
645 421 2Ɛ1
657 431 2ƐƐ
661 433 301
667 439 307
701 449 315
711 457 321
715 461 325
717 463 327
723 467 32Ɛ
737 479 33Ɛ
747 487 347
753 491 34Ɛ
763 499 357
767 503 35Ɛ
775 509 365
1011 521 375

Prime factors
n Oct Dec Duo
1 1 1 1
2 2 2 2
3 3 3 3
4 2 x 2 (22) 2 x 2 (22) 2 x 2 (22)
5 5 5 5
6 2 x 3 2 x 3 2 x 3
7 7 7 7
10 (8) (8) 2 x 2 x 2 (23) 2 x 2 x 2 (23) 2 x 2 x 2 (23)
11 (9) (9) 3 x 3 (32) 3 x 3 (32) 3 x 3 (32)
12 (10) (ᘔ) 2 x 5 2 x 5 2 x 5
13 (11) (Ɛ) 13 11 Ɛ
14 (12) (10) 2 x 2 x 3 (22 x 3) 2 x 2 x 3 (22 x 3) 2 x 2 x 3 (22 x 3)
15 (13) (11) 15 13 11
16 (14) (12) 2 x 7 2 x 7 2 x 7
17 (15) (13) 3 x 5 3 x 5 3 x 5
20 (16) (14) 2 x 2 x 2 x 2 (24) 2 x 2 x 2 x 2 (24) 2 x 2 x 2 x 2 (24)
21 (17) (15) 21 17 15
22 (18) (16) 2 x 3 x 3 (2 x 32) 2 x 3 x 3 (2 x 32) 2 x 3 x 3 (2 x 32)
23 (19) (17) 23 19 17
24 (20) (18) 2 x 2 x 5 (22 x 5) 2 x 2 x 5 (22 x 5) 2 x 2 x 5 (22 x 5)
25 (21) (19) 3 x 7 3 x 7 3 x 7
26 (22) (1ᘔ) 2 x 13 2 x 11 2 x Ɛ
27 (23) (1Ɛ) 27 23
30 (24) (20) 2 x 2 x 2 x 3 (23 x 3) 2 x 2 x 2 x 3 (23 x 3) 2 x 2 x 2 x 3 (23 x 3)
31 (25) (21) 5 x 5 (52) 5 x 5 (52) 5 x 5 (52)
32 (26) (22) 2 x 15 2 x 13 2 x 11
33 (27) (23) 3 x 3 x 3 (33) 3 x 3 x 3 (33) 3 x 3 x 3 (33)
34 (28) (24) 2 x 2 x 7 (22 x 7) 2 x 2 x 7 (22 x 7) 2 x 2 x 7 (22 x 7)
35 (29) (25) 35 29 25
36 (30) (26) 2 x 3 x 5 2 x 3 x 5 2 x 3 x 5
37 (31) (27) 37 31 27
40 (32) (28) 2 x 2 x 2 x 2 x 2 (25) 2 x 2 x 2 x 2 x 2 (25) 2 x 2 x 2 x 2 x 2 (25)
41 (33) (29) 3 x 13 3 x 11 3 x Ɛ
42 (34) (2ᘔ) 2 x 21 2 x 17 2 x 15
43 (35) (2Ɛ) 5 x 7 5 x 7 5 x 7
44 (36) (30) 2 x 2 x 3 x 3 (22 x 32) 2 x 2 x 3 x 3 (22 x 32) 2 x 2 x 3 x 3 (22 x 32)
45 (37) (31) 45 37 31
46 (38) (32) 2 x 23 2 x 19 2 x 17
47 (39) (33) 3 x 15 3 x 13 3 x 11
50 (40) (34) 2 x 2 x 2 x 5 (23 x 5) 2 x 2 x 2 x 5 (23 x 5) 2 x 2 x 2 x 5 (23 x 5)
51 (41) (35) 51 41 35
52 (42) (36) 2 x 3 x 7 2 x 3 x 7 2 x 3 x 7
53 (43) (37) 53 43 37
54 (44) (38) 2 x 2 x 13 (22 x 13) 2 x 2 x 11 (22 x 11) 2 x 2 x Ɛ (22 x Ɛ)
55 (45) (39) 3 x 3 x 5 (32 x 5) 3 x 3 x 5 (32 x 5) 3 x 3 x 5 (32 x 5)
56 (46) (3ᘔ) 2 x 27 2 x 23 2 x 1Ɛ
57 (47) (3Ɛ) 57 47
60 (48) (40) 2 x 2 x 2 x 2 x 3 (24 x 3) 2 x 2 x 2 x 2 x 3 (24 x 3) 2 x 2 x 2 x 2 x 3 (24 x 3)
61 (49) (41) 7 x 7 (72) 7 x 7 (72) 7 x 7 (72)
62 (50) (42) 2 x 5 x 5 (2 x 52) 2 x 5 x 5 (2 x 52) 2 x 5 x 5 (2 x 52)
63 (51) (43) 3 x 21 3 x 17 3 x 15
64 (52) (44) 2 x 2 x 15 (22 x 15) 2 x 2 x 13 (22 x 13) 2 x 2 x 11 (22 x 11)
65 (53) (45) 65 53 45
66 (54) (46) 2 x 3 x 3 x 3 (2 x 33) 2 x 3 x 3 x 3 (2 x 33) 2 x 3 x 3 x 3 (2 x 33)
67 (55) (47) 5 x 13 5 x 11 5 x Ɛ
70 (56) (48) 2 x 2 x 2 x 7 (23 x 7) 2 x 2 x 2 x 7 (23 x 7) 2 x 2 x 2 x 7 (23 x 7)
71 (57) (49) 3 x 23 3 x 19 3 x 17
72 (58) (4ᘔ) 2 x 35 2 x 29 2 x 25
73 (59) (4Ɛ) 73 59
74 (60) (50) 2 x 2 x 3 x 5 (22 x 3 x 5) 2 x 2 x 3 x 5 (22 x 3 x 5) 2 x 2 x 3 x 5 (22 x 3 x 5)
75 (61) (51) 75 61 51
76 (62) (52) 2 x 37 2 x 31 2 x 27
77 (63) (53) 3 x 3 x 7 (32 x 7) 3 x 3 x 7 (32 x 7) 3 x 3 x 7 (32 x 7)
100 (64) (54) 2 x 2 x 2 x 2 x 2 x 2 (26) 2 x 2 x 2 x 2 x 2 x 2 (26) 2 x 2 x 2 x 2 x 2 x 2 (26)
101 (65) (55) 5 x 15 5 x 13 5 x 11
102 (66) (56) 2 x 3 x 13 2 x 3 x 11 2 x 3 x Ɛ
103 (67) (57) 103 67 57
104 (68) (58) 2 x 2 x 21 (22 x 21) 2 x 2 x 17 (22 x 17) 2 x 2 x 15 (22 x 15)
105 (69) (59) 3 x 27 3 x 23 3 x 1Ɛ
106 (70) (5ᘔ) 2 x 5 x 7 2 x 5 x 7 2 x 5 x 7
107 (71) (5Ɛ) 107 71
110 (72) (60) 2 x 2 x 2 x 3 x 3 (23 x 32) 2 x 2 x 2 x 3 x 3 (23 x 32) 2 x 2 x 2 x 3 x 3 (23 x 32)
111 (73) (61) 111 73 61
112 (74) (62) 2 x 45 2 x 37 2 x 31
113 (75) (63) 3 x 5 x 5 (3 x 52) 3 x 5 x 5 (3 x 52) 3 x 5 x 5 (3 x 52)
114 (76) (64) 2 x 2 x 23 (22 x 23) 2 x 2 x 19 (22 x 19) 2 x 2 x 17 (22 x 17)
115 (77) (65) 7 x 13 7 x 11 7 x Ɛ
116 (78) (66) 2 x 3 x 15 2 x 3 x 13 2 x 3 x 11
117 (79) (67) 117 79 67
120 (80) (68) 2 x 2 x 2 x 2 x 5 (24 x 5) 2 x 2 x 2 x 2 x 5 (24 x 5) 2 x 2 x 2 x 2 x 5 (24 x 5)
121 (81) (69) 3 x 3 x 3 x 3 (34) 3 x 3 x 3 x 3 (34) 3 x 3 x 3 x 3 (34)
122 (82) (6ᘔ) 2 x 51 2 x 41 2 x 35
123 (83) (6Ɛ) 83 83
124 (84) (70) 2 x 2 x 3 x 7 (22 x 3 x 7) 2 x 2 x 3 x 7 (22 x 3 x 7) 2 x 2 x 3 x 7 (22 x 3 x 7)
125 (85) (71) 5 x 21 5 x 17 5 x 15
126 (86) (72) 2 x 53 2 x 43 2 x 37
127 (87) (73) 3 x 35 3 x 29 3 x 25
130 (88) (74) 2 x 2 x 2 x 13 (23 x 13) 2 x 2 x 2 x 11 (23 x 11) 2 x 2 x 2 x Ɛ (23 x Ɛ)
131 (89) (75) 131 89 75
132 (90) (76) 2 x 3 x 3 x 5 (2 x 32 x 5) 2 x 3 x 3 x 5 (2 x 32 x 5) 2 x 3 x 3 x 5 (2 x 32 x 5)
133 (91) (77) 7 x 15 7 x 13 7 x 11
134 (92) (78) 2 x 2 x 27 (22 x 27) 2 x 2 x 23 (22 x 23) 2 x 2 x 1Ɛ (22 x 1Ɛ)
135 (93) (79) 3 x 37 3 x 31 3 x 27
136 (94) (7ᘔ) 2 x 57 2 x 47 2 x 3Ɛ
137 (95) (7Ɛ) 5 x 23 5 x 19 5 x 17
140 (96) (80) 2 x 2 x 2 x 2 x 2 x 3 (25 x 3) 2 x 2 x 2 x 2 x 2 x 3 (25 x 3) 2 x 2 x 2 x 2 x 2 x 3 (25 x 3)
141 (97) (81) 141 97 81
142 (98) (82) 2 x 7 x 7 (2 x 72) 2 x 7 x 7 (2 x 72) 2 x 7 x 7 (2 x 72)
143 (99) (83) 3 x 3 x 13 (32 x 13) 3 x 3 x 11 (32 x 11) 3 x 3 x Ɛ (32 x Ɛ)
144 (100) (84) 2 x 2 x 5 x 5 (22 x 52) 2 x 2 x 5 x 5 (22 x 52) 2 x 2 x 5 x 5 (22 x 52)
145 (101) (85) 145 101 85
146 (102) (86) 2 x 3 x 21 2 x 3 x 17 2 x 3 x 15
147 (103) (87) 147 103 87
150 (104) (88) 2 x 2 x 2 x 15 (23 x 15) 2 x 2 x 2 x 13 (23 x 13) 2 x 2 x 2 x 11 (23 x 11)
151 (105) (89) 3 x 5 x 7 3 x 5 x 7 3 x 5 x 7
152 (106) (8ᘔ) 2 x 65 2 x 53 2 x 45
153 (107) (8Ɛ) 153 107
154 (108) (90) 2 x 2 x 3 x 3 x 3 (22 x 33) 2 x 2 x 3 x 3 x 3 (22 x 33) 2 x 2 x 3 x 3 x 3 (22 x 33)
155 (109) (91) 155 109 91
156 (110) (92) 2 x 5 x 13 2 x 5 x 11 2 x 5 x Ɛ
157 (111) (93) 3 x 45 3 x 37 3 x 31
160 (112) (94) 2 x 2 x 2 x 2 x 7 (24 x 7) 2 x 2 x 2 x 2 x 7 (24 x 7) 2 x 2 x 2 x 2 x 7 (24 x 7)
161 (113) (95) 161 113 95
162 (114) (96) 2 x 3 x 23 2 x 3 x 19 2 x 3 x 17
163 (115) (97) 5 x 27 5 x 23 5 x 1Ɛ
164 (116) (98) 2 x 2 x 35 (22 x 35) 2 x 2 x 29 (22 x 29) 2 x 2 x 25 (22 x 25)
165 (117) (99) 3 x 3 x 15 (33 x 15) 3 x 3 x 13 (33 x 13) 3 x 3 x 11 (33 x 11)
166 (118) (9ᘔ) 2 x 73 2 x 59 2 x 4Ɛ
167 (119) (9Ɛ) 7 x 21 7 x 17 7 x 15
170 (120) (ᘔ0) 2 x 2 x 2 x 3 x 5 (23 x 3 x 5) 2 x 2 x 2 x 3 x 5 (23 x 3 x 5) 2 x 2 x 2 x 3 x 5 (23 x 3 x 5)
171 (121) (ᘔ1) 13 x 13 11 x 11 Ɛ x Ɛ
172 (122) (ᘔ2) 2 x 75 2 x 61 2 x 51
173 (123) (ᘔ3) 3 x 51 3 x 41 3 x 35
174 (124) (ᘔ4) 2 x 2 x 37 (22 x 37) 2 x 2 x 31 (22 x 31) 2 x 2 x 27 (22 x 27)
175 (125) (ᘔ5) 5 x 5 x 5 (53) 5 x 5 x 5 (53) 5 x 5 x 5 (53)
176 (126) (ᘔ6) 2 x 3 x 3 x 7 (2 x 32 x 7) 2 x 3 x 3 x 7 (2 x 32 x 7) 2 x 3 x 3 x 7 (2 x 32 x 7)
177 (127) (ᘔ7) 177 127 ᘔ7
200 (128) (ᘔ8) 2 x 2 x 2 x 2 x 2 x 2 x 2 (27) 2 x 2 x 2 x 2 x 2 x 2 x 2 (27) 2 x 2 x 2 x 2 x 2 x 2 x 2 (27)
201 (129) (ᘔ9) 3 x 53 3 x 43 3 x 37
202 (130) (ᘔᘔ) 2 x 5 x 15 2 x 5 x 13 2 x 5 x 11
203 (131) (ᘔƐ) 203 131 ᘔƐ
204 (132) (Ɛ0) 2 x 2 x 3 x 13 (22 x 3 x 13) 2 x 2 x 3 x 11 (22 x 3 x 11) 2 x 2 x 3 x Ɛ (22 x 3 x Ɛ)
205 (133) (Ɛ1) 7 x 23 7 x 19 7 x 17
206 (134) (Ɛ2) 2 x 103 2 x 67 2 x 57
207 (135) (Ɛ3) 3 x 3 x 3 x 5 (33 x 5) 3 x 3 x 3 x 5 (33 x 5) 3 x 3 x 3 x 5 (33 x 5)
210 (136) (Ɛ4) 2 x 2 x 2 x 21 (23 x 21) 2 x 2 x 2 x 17 (23 x 17) 2 x 2 x 2 x 15 (23 x 15)
211 (137) (Ɛ5) 211 137 Ɛ5
212 (138) (Ɛ6) 2 x 3 x 27 2 x 3 x 23 2 x 3 x 1Ɛ
213 (139) (Ɛ7) 213 139 Ɛ7
214 (140) (Ɛ8) 2 x 2 x 5 x 7 (22 x 5 x 7) 2 x 2 x 5 x 7 (22 x 5 x 7) 2 x 2 x 5 x 7 (22 x 5 x 7)
215 (141) (Ɛ9) 3 x 57 3 x 47 3 x 3Ɛ
216 (142) (Ɛᘔ) 2 x 107 2 x 71 2 x 5Ɛ
217 (143) (ƐƐ) 13 x 15 11 x 13 Ɛ x 11
220 (144) (100) 2 x 2 x 2 x 2 x 3 x 3 (24 x 32) 2 x 2 x 2 x 2 x 3 x 3 (24 x 32) 2 x 2 x 2 x 2 x 3 x 3 (24 x 32)

Smith numbers
Oct Dec Duo
4 4 4
17 22 18
56 27 19
64 58 22
71 85 28
125 94 73
... 121 76
... 166 ᘔ5
... 202 Ɛ3
... 265 Ɛ6

Findings


Fibonacci

Fibonacci numbers
Oct Dec Duo
0 0 0
1 1 1
1 1 1
2 2 2
3 3 3
5 5 5
10 8 8
15 13 11
25 21 19
42 34 2ᘔ
67 55 47
131 89 75
220 144 100
351 233 175
571 377 275
1142 610 42ᘔ

Repfigit numbers
Oct Dec Duo
10 14 11
13 19 15
17 28
20 47 22
26 61 2ᘔ
30 75 31
40 197 33
45 742 44
50 1104 49
60 1537 55
70 2208 62
73 2580 66
134 3684 77
... 4788 88
... 7385 99
... 7647 ᘔᘔ
... 7909 ƐƐ
... 3 1331 125
... 3 4285 ...
... 3 4348 ...
... 5 5604 ...
... 6 2662 ...
... 8 6935 ...
... 9 3993 ...
... 12 0284 ...
... 12 9106 ...
... 14 7640 ...
... 15 6146 ...
... 17 4680 ...
... 18 3186 ...
... 29 8320 ...
... 35 5419 ...
... 69 4280 ...
... 92 5993 ...
... 108 4051 ...
... 791 3837 ...
... 1143 6171 ...
... 3344 5755 ...
... 4412 1607 ...
... 1 2957 2008 ...
... 2 5113 3297 ...

Repfigit number clusters
Oct Dec Duo
(10, 20, 30, 40, 50, 60, 70) (14, 28) (11, 22, 33, 44, 55, 66, 77, 88, 99, ᘔᘔ, ƐƐ)
(13, 26) (1104, 2208) (15, 2ᘔ)
(20, 40, 60) (3 1331, 6 2662, 9 3993) (22, 44, 66, 88, ᘔᘔ)
(30, 60) ... (31, 62)
... ... (33, 66, 99)
... ... (44, 88)
... ... (55, ᘔᘔ)

Findings


Factorials

n Oct Dec Duo
0 1 1 1
1 1 1 1
2 2 2 2
3 6 6 6
4 30 24 20
5 170 120 ᘔ0
6 1320 720 500
7 1 1660 5040 2Ɛ00
10 (8) (8) 11 6600 4 0320 1 Ɛ400
11 (9) (9) 130 4600 36 2880 15 6000
12 (10) (ᘔ) 1565 7400 362 8800 127 0000
13 (11) (Ɛ) 2 3021 2400 3991 6800 1145 0000
14 (12) (10) 34 4317 6000 4 7900 1600 1 1450 0000
15 (13) (11) 563 1214 6000 62 2702 0800 12 5950 0000
16 (14) (12) 1 2114 1662 4000 871 7829 1200 14ᘔ 8Ɛᘔ0 0000
17 (15) (13) 23 0167 3565 4000 1 3076 7436 8000 1915 2960 0000
20 (16) (14) 460 3567 3530 0000 20 9227 8988 8000 2 41ᘔƐ 8800 0000
21 (17) (15) 1 2067 7356 6330 0000 355 6874 2809 6000 33 ᘔ867 3400 0000
22 (18) (16) 26 5756 6312 3460 0000 6402 3737 0572 8000 4Ɛᘔ 09ᘔƐ 0000 0000
23 (19) (17) 660 1271 1406 4220 0000 12 1645 1004 0883 2000 7ᘔ8Ɛ 3835 0000 0000
24 (20) (18) 2 0703 3167 6202 5500 0000 243 2902 0081 7664 0000 11 1ᘔᘔᘔ 1984 0000 0000

Findings


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