Advertisements

## About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , , . Bookmark the permalink.

### 5 Responses to Patterns| Base-10 integers in pair bases b1 and b2

1. Paul says:

In terms of the specific pattern where we have two consecutive sets of 2 numbers and the 5th number is the last number plus 5 then those are the only ones I can find with bases from 2 to 21 and the integer up to 10 million. Also there are no more number with those 2 bases up to 100 million. Now if we relax the pattern match a little such that the match is a single number we find quite a few more, in the same range b1<20 && b2<21 and N <=10000000 there are just over 1000 examples, Do you want them all or just a few?

Paul.

• benvitalis says:

Just a few examples will do

• benvitalis says:

Very Nice. I’ve only checked up to 2 million numbers. Initially, I started with base-9: a*9^3 + b*9^2 + c*9 + d, base-5: a*5^4 + b*5^3 + c*5^2 + d*5 + e

2. Paul says:

ok here’s a snapshot
N b1 b2 b-1 b2
2180306 , 5 , 8 , 1024232211 , 10242322
2196768 , 5 , 8 , 1030244033 , 10302440
2196996 , 5 , 8 , 1030300441 , 10303004
2197000 , 5 , 8 , 1030301000 , 10303010
2364049 , 5 , 8 , 1101122144 , 11011221
2364050 , 5 , 8 , 1101122200 , 11011222
4561049 , 5 , 8 , 2131423144 , 21314231
4561050 , 5 , 8 , 2131423200 , 21314232
4562971 , 5 , 8 , 2132003341 , 21320033
4728099 , 5 , 8 , 2202244344 , 22022443
4728100 , 5 , 8 , 2202244400 , 22022444
4728331 , 5 , 8 , 2202301311 , 22023013
6927049 , 5 , 8 , 3233131144 , 32331311
6927050 , 5 , 8 , 3233131200 , 32331312
7092384 , 5 , 8 , 3303424014 , 33034240
7094306 , 5 , 8 , 3304004211 , 33040042
7110793 , 5 , 8 , 3310021133 , 33100211
9291331 , 5 , 8 , 4334310311 , 43343103
9307793 , 5 , 8 , 4340322133 , 43403221
9474843 , 5 , 8 , 4411143333 , 44111433
9475074 , 5 , 8 , 4411200244 , 44112002
9475075 , 5 , 8 , 4411200300 , 44112003

1343 , 6 , 11 , 10115 , 1011
1344 , 6 , 11 , 10120 , 1012
2687 , 6 , 11 , 20235 , 2023
2688 , 6 , 11 , 20240 , 2024
4031 , 6 , 11 , 30355 , 3035
4037 , 6 , 11 , 30405 , 3040
4038 , 6 , 11 , 30410 , 3041
4115 , 6 , 11 , 31015 , 3101
4116 , 6 , 11 , 31020 , 3102
5381 , 6 , 11 , 40525 , 4052
5382 , 6 , 11 , 40530 , 4053
5459 , 6 , 11 , 41135 , 4113
5460 , 6 , 11 , 41140 , 4114
6803 , 6 , 11 , 51255 , 5125
6809 , 6 , 11 , 51305 , 5130
6810 , 6 , 11 , 51310 , 5131
1806335 , 6 , 11 , 102414355 , 1024143
1806336 , 6 , 11 , 102414400 , 1024144
3620730 , 6 , 11 , 205334350 , 2053343
3725506 , 6 , 11 , 211503414 , 2115034
3733503 , 6 , 11 , 212004423 , 2120044
5539901 , 6 , 11 , 314423405 , 3144234
5643982 , 6 , 11 , 320545314 , 3205453
5652035 , 6 , 11 , 321050455 , 3210504
5652036 , 6 , 11 , 321050500 , 3210505
7459071 , 6 , 11 , 423512423 , 4235124
7467068 , 6 , 11 , 424013432 , 4240134
7571149 , 6 , 11 , 430135341 , 4301353
9377547 , 6 , 11 , 532554323 , 5325543
9385544 , 6 , 11 , 533055332 , 5330553
9386238 , 6 , 11 , 533102450 , 5331024

363 , 7 , 19 , 1026 , 102
364 , 7 , 19 , 1030 , 103
727 , 7 , 19 , 2056 , 205
728 , 7 , 19 , 2060 , 206
741 , 7 , 19 , 2106 , 210
742 , 7 , 19 , 2110 , 211
1105 , 7 , 19 , 3136 , 313
1106 , 7 , 19 , 3140 , 314
1469 , 7 , 19 , 4166 , 416
1483 , 7 , 19 , 4216 , 421
1484 , 7 , 19 , 4220 , 422
1847 , 7 , 19 , 5246 , 524
1848 , 7 , 19 , 5250 , 525
1862 , 7 , 19 , 5300 , 530
2225 , 7 , 19 , 6326 , 632
2226 , 7 , 19 , 6330 , 633
132508 , 7 , 19 , 1061215 , 10612
137579 , 7 , 19 , 1112051 , 11120
275159 , 7 , 19 , 2224133 , 22241
412739 , 7 , 19 , 3336215 , 33362
412984 , 7 , 19 , 3340015 , 33400
550563 , 7 , 19 , 4452066 , 44520
550564 , 7 , 19 , 4452100 , 44521
555623 , 7 , 19 , 4502615 , 45026
688143 , 7 , 19 , 5564151 , 55641
693202 , 7 , 19 , 5614666 , 56146
693215 , 7 , 19 , 5615015 , 56150

17 , 4 , 17 , 101 , 10
34 , 4 , 17 , 202 , 20
51 , 4 , 17 , 303 , 30
52 , 4 , 17 , 310 , 31
291 , 4 , 17 , 10203 , 102
596 , 4 , 17 , 21110 , 211
887 , 4 , 17 , 31313 , 313
901 , 4 , 17 , 32011 , 320
4965 , 4 , 17 , 1031211 , 1031
5203 , 4 , 17 , 1101103 , 1101
10168 , 4 , 17 , 2132320 , 2132
10406 , 4 , 17 , 2202212 , 2202
15609 , 4 , 17 , 3303321 , 3303
15623 , 4 , 17 , 3310013 , 3310
88759 , 4 , 17 , 111222313 , 11122
177754 , 4 , 17 , 223121122 , 22312
181800 , 4 , 17 , 230120220 , 23012
1518153 , 4 , 17 , 11302221021 , 113022
1586951 , 4 , 17 , 12003130013 , 120031
3105339 , 4 , 17 , 23312020323 , 233120

Paul.