examples of uniqueness of digital representation of small numbers in a few bases

While a string of digits is not unique to any base $b$, within each base $b$ the base $b$ representation of an integer $n$ is unique. The following table gives the base 2 (binary) through base 10 (decimal) representations of $n$ from 1 to 50. It will be seen how “10” occurs in several cells diagonally in the table, but in each column it corresponds to the case $n=b$ only. Likewise, “11” also occurs diagonally, but in each column it corresponds to $n=b+1$ only. One other generality to mention: the representation $n$ is the same for all bases $b>n$.

$b=2$	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1
10	2	2	2	2	2	2	2	2
11	10	3	3	3	3	3	3	3
100	11	10	4	4	4	4	4	4
101	12	11	10	5	5	5	5	5
110	20	12	11	10	6	6	6	6
111	21	13	12	11	10	7	7	7
1000	22	20	13	12	11	10	8	8
1001	100	21	14	13	12	11	10	9
1010	101	22	20	14	13	12	11	10
1011	102	23	21	15	14	13	12	11
1100	110	30	22	20	15	14	13	12
1101	111	31	23	21	16	15	14	13
1110	112	32	24	22	20	16	15	14
1111	120	33	30	23	21	17	16	15
10000	121	100	31	24	22	20	17	16
10001	122	101	32	25	23	21	18	17
10010	200	102	33	30	24	22	20	18
10011	201	103	34	31	25	23	21	19
10100	202	110	40	32	26	24	22	20
10101	210	111	41	33	30	25	23	21
10110	211	112	42	34	31	26	24	22
10111	212	113	43	35	32	27	25	23
11000	220	120	44	40	33	30	26	24
11001	221	121	100	41	34	31	27	25
11010	222	122	101	42	35	32	28	26
11011	1000	123	102	43	36	33	30	27
11100	1001	130	103	44	40	34	31	28
11101	1002	131	104	45	41	35	32	29
11110	1010	132	110	50	42	36	33	30
11111	1011	133	111	51	43	37	34	31
100000	1012	200	112	52	44	40	35	32
100001	1020	201	113	53	45	41	36	33
100010	1021	202	114	54	46	42	37	34
100011	1022	203	120	55	50	43	38	35
100100	1100	210	121	100	51	44	40	36
100101	1101	211	122	101	52	45	41	37
100110	1102	212	123	102	53	46	42	38
100111	1110	213	124	103	54	47	43	39
101000	1111	220	130	104	55	50	44	40
101001	1112	221	131	105	56	51	45	41
101010	1120	222	132	110	60	52	46	42
101011	1121	223	133	111	61	53	47	43
101100	1122	230	134	112	62	54	48	44
101101	1200	231	140	113	63	55	50	45
101110	1201	232	141	114	64	56	51	46
101111	1202	233	142	115	65	57	52	47
110000	1210	300	143	120	66	60	53	48
110001	1211	301	144	121	100	61	54	49
110010	1212	302	200	122	101	62	55	50

Title	examples of uniqueness of digital representation of small numbers in a few bases
Canonical name	ExamplesOfUniquenessOfDigitalRepresentationOfSmallNumbersInAFewBases
Date of creation	2013-03-22 18:52:25
Last modified on	2013-03-22 18:52:25
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 11A63
Classification	msc 11A05