Duodecimal

The duodecimal system (also known as base 12 or dozenal) is a positional notation numeral system using twelve as its base.

Unicode points U+218C and U+218D seem to be reserved for the Dwiggins digits (stylized X and E).

Powers

		n±		n±
	n±
		0		3	9
		zero		three	nine
	n±
		2	4	1
		two	four	one

Multiplication table

Duodecimal multiplication table

×	0	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10	11	12	13	14	15	16	17	18	19	1ᘔ	1Ɛ	20
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10	11	12	13	14	15	16	17	18	19	1ᘔ	1Ɛ	20
2	0	2	4	6	8	ᘔ	10	12	14	16	18	1ᘔ	20	22	24	26	28	2ᘔ	30	32	34	36	38	3ᘔ	40
3	0	3	6	9	10	13	16	19	20	23	26	29	30	33	36	39	40	43	46	49	50	53	56	59	60
4	0	4	8	10	14	18	20	24	28	30	34	38	40	44	48	50	54	58	60	64	68	70	74	78	80
5	0	5	ᘔ	13	18	21	26	2Ɛ	34	39	42	47	50	55	5ᘔ	63	68	71	76	7Ɛ	84	89	92	97	ᘔ0
6	0	6	10	16	20	26	30	36	40	46	50	56	60	66	70	76	80	86	90	96	ᘔ0	ᘔ6	Ɛ0	Ɛ6	100
7	0	7	12	19	24	2Ɛ	36	41	48	53	5ᘔ	65	70	77	82	89	94	9Ɛ	ᘔ6	Ɛ1	Ɛ8	103	10ᘔ	115	120
8	0	8	14	20	28	34	40	48	54	60	68	74	80	88	94	ᘔ0	ᘔ8	Ɛ4	100	108	114	120	128	134	140
9	0	9	16	23	30	39	46	53	60	69	76	83	90	99	ᘔ6	Ɛ3	100	109	116	123	130	139	146	153	160
ᘔ	0	ᘔ	18	26	34	42	50	5ᘔ	68	76	84	92	ᘔ0	ᘔᘔ	Ɛ8	106	114	122	130	13ᘔ	148	156	164	172	180
Ɛ	0	Ɛ	1ᘔ	29	38	47	56	65	74	83	92	ᘔ1	Ɛ0	ƐƐ	10ᘔ	119	128	137	146	155	164	173	182	191	1ᘔ0
10	0	10	20	30	40	50	60	70	80	90	ᘔ0	Ɛ0	100	110	120	130	140	150	160	170	180	190	1ᘔ0	1Ɛ0	200
11	0	11	22	33	44	55	66	77	88	99	ᘔᘔ	ƐƐ	110	121	132	143	154	165	176	187	198	1ᘔ9	1Ɛᘔ	20Ɛ	220
12	0	12	24	36	48	5ᘔ	70	82	94	ᘔ6	Ɛ8	10ᘔ	120	132	144	156	168	17ᘔ	190	1ᘔ2	1Ɛ4	206	218	22ᘔ	240
13	0	13	26	39	50	63	76	89	ᘔ0	Ɛ3	106	119	130	143	156	169	180	193	1ᘔ6	1Ɛ9	210	223	236	249	260
14	0	14	28	40	54	68	80	94	ᘔ8	100	114	128	140	154	168	180	194	1ᘔ8	200	214	228	240	254	268	280
15	0	15	2ᘔ	43	58	71	86	9Ɛ	Ɛ4	109	122	137	150	165	17ᘔ	193	1ᘔ8	201	216	22Ɛ	244	259	272	287	2ᘔ0
16	0	16	30	46	60	76	90	ᘔ6	100	116	130	146	160	176	190	1ᘔ6	200	216	230	246	260	276	290	2ᘔ6	300
17	0	17	32	49	64	7Ɛ	96	Ɛ1	108	123	13ᘔ	155	170	187	1ᘔ2	1Ɛ9	214	22Ɛ	246	261	278	293	2ᘔᘔ	305	320
18	0	18	34	50	68	84	ᘔ0	Ɛ8	114	130	148	164	180	198	1Ɛ4	210	228	244	260	278	294	2Ɛ0	308	324	340
19	0	19	36	53	70	89	ᘔ6	103	120	139	156	173	190	1ᘔ9	206	223	240	259	276	293	2Ɛ0	309	326	343	360
1ᘔ	0	1ᘔ	38	56	74	92	Ɛ0	10ᘔ	128	146	164	182	1ᘔ0	1Ɛᘔ	218	236	254	272	290	2ᘔᘔ	308	326	344	362	380
1Ɛ	0	1Ɛ	3ᘔ	59	78	97	Ɛ6	115	134	153	172	191	1Ɛ0	20Ɛ	22ᘔ	249	268	287	2ᘔ6	305	324	343	362	381	3ᘔ0
20	0	20	40	60	80	ᘔ0	100	120	140	160	180	1ᘔ0	200	220	240	260	280	2ᘔ0	300	320	340	360	380	3ᘔ0	400

Conversion tables to and from decimal

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0.01 and ƐƐƐ,ƐƐƐ.ƐƐ to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:

123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08

This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get:

(duodecimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.583333333333... + 0.055555555555...

Now, because the summands are already converted to base ten, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result:

Duodecimal  ----->  Decimal

  100,000     =    248,832
   20,000     =     41,472
    3,000     =      5,184
      400     =        576
       50     =         60
 +      6     =   +      6
        0.7   =          0.583333333333...
        0.08  =          0.055555555555...
--------------------------------------------
  123,456.78  =    296,130.638888888888...

That is, (duodecimal) 123,456.78 equals (decimal) 296,130.638 ≈ 296,130.64

If the given number is in decimal and the target base is duodecimal, the method is basically same. Using the digit conversion tables:

(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,ᘔ54 + Ɛ,6ᘔ8 + 1,8ᘔ0 + 294 + 42 + 6 + 0.849724972497249724972497... + 0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62...

However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in duodecimal as well. In decimal, 6 + 6 equals 12, but in duodecimal it equals 10; so, if using decimal arithmetic with duodecimal numbers one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result:

  Decimal  ----->  Duodecimal

  100,000     =     49,ᘔ54
   20,000     =      Ɛ,6ᘔ8
    3,000     =      1,8ᘔ0
      400     =        294
       50     =         42
 +      6     =   +      6
        0.7   =          0.849724972497249724972497...
        0.08  =          0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62...
--------------------------------------------------------
  123,456.78  =     5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ05915343ᘔ...

That is, (decimal) 123,456.78 equals (duodecimal) 5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ059153... ≈ 5Ɛ,540.94

Duodecimal to decimal digit conversion

Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.
1,000,000	2,985,984	100,000	248,832	10,000	20,736	1,000	1,728	100	144	10	12	1	1	0.1	0.083	0.01	0.00694
2,000,000	5,971,968	200,000	497,664	20,000	41,472	2,000	3,456	200	288	20	24	2	2	0.2	0.16	0.02	0.0138
3,000,000	8,957,952	300,000	746,496	30,000	62,208	3,000	5,184	300	432	30	36	3	3	0.3	0.25	0.03	0.02083
4,000,000	11,943,936	400,000	995,328	40,000	82,944	4,000	6,912	400	576	40	48	4	4	0.4	0.3	0.04	0.027
5,000,000	14,929,920	500,000	1,244,160	50,000	103,680	5,000	8,640	500	720	50	60	5	5	0.5	0.416	0.05	0.03472
6,000,000	17,915,904	600,000	1,492,992	60,000	124,416	6,000	10,368	600	864	60	72	6	6	0.6	0.5	0.06	0.0416
7,000,000	20,901,888	700,000	1,741,824	70,000	145,152	7,000	12,096	700	1,008	70	84	7	7	0.7	0.583	0.07	0.04861
8,000,000	23,887,872	800,000	1,990,656	80,000	165,888	8,000	13,824	800	1,152	80	96	8	8	0.8	0.6	0.08	0.05
9,000,000	26,873,856	900,000	2,239,488	90,000	186,624	9,000	15,552	900	1,296	90	108	9	9	0.9	0.75	0.09	0.0625
ᘔ,000,000	29,859,840	ᘔ00,000	2,488,320	ᘔ0,000	207,360	ᘔ,000	17,280	ᘔ00	1,440	ᘔ0	120	ᘔ	10	0.ᘔ	0.83	0.0ᘔ	0.0694
Ɛ,000,000	32,845,824	Ɛ00,000	2,737,152	Ɛ0,000	228,096	Ɛ,000	19,008	Ɛ00	1,584	Ɛ0	132	Ɛ	11	0.Ɛ	0.916	0.0Ɛ	0.07638

Decimal to duodecimal digit conversion

Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.
100,000	49,ᘔ54	10,000	5,954	1,000	6Ɛ4	100	84	10	ᘔ	1	1	0.1	0.12497	0.01	0.015343ᘔ0Ɛ62ᘔ68781Ɛ059
200,000	97,8ᘔ8	20,000	Ɛ,6ᘔ8	2,000	1,1ᘔ8	200	148	20	18	2	2	0.2	0.2497	0.02	0.02ᘔ68781Ɛ05915343ᘔ0Ɛ6
300,000	125,740	30,000	15,440	3,000	1,8ᘔ0	300	210	30	26	3	3	0.3	0.37249	0.03	0.043ᘔ0Ɛ62ᘔ68781Ɛ059153
400,000	173,594	40,000	1Ɛ,194	4,000	2,394	400	294	40	34	4	4	0.4	0.4972	0.04	0.05915343ᘔ0Ɛ62ᘔ68781Ɛ
500,000	201,428	50,000	24,Ɛ28	5,000	2,ᘔ88	500	358	50	42	5	5	0.5	0.6	0.05	0.07249
600,000	24Ɛ,280	60,000	2ᘔ,880	6,000	3,580	600	420	60	50	6	6	0.6	0.7249	0.06	0.08781Ɛ05915343ᘔ0Ɛ62ᘔ6
700,000	299,114	70,000	34,614	7,000	4,074	700	4ᘔ4	70	5ᘔ	7	7	0.7	0.84972	0.07	0.0ᘔ0Ɛ62ᘔ68781Ɛ05915343
800,000	326,Ɛ68	80,000	3ᘔ,368	8,000	4,768	800	568	80	68	8	8	0.8	0.9724	0.08	0.0Ɛ62ᘔ68781Ɛ05915343ᘔ
900,000	374,ᘔ00	90,000	44,100	9,000	5,260	900	630	90	76	9	9	0.9	0.ᘔ9724	0.09	0.10Ɛ62ᘔ68781Ɛ05915343ᘔ

Conversion of powers

Exponent	b=2		b=3		b=4		b=5		b=6		b=7
	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.
b6	64	54	729	509	4,096	2,454	15,625	9,061	46,656	23,000	117,649	58,101
b5	32	28	243	183	1,024	714	3,125	1,985	7,776	4,600	16,807	9,887
b4	16	14	81	69	256	194	625	441	1,296	900	2,401	1,481
b3	8	8	27	23	64	54	125	ᘔ5	216	160	343	247
b2	4	4	9	9	16	14	25	21	36	30	49	41
b1	2	2	3	3	4	4	5	5	6	6	7	7
b−1	0.5	0.6	0.3	0.4	0.25	0.3	0.2	0.2497	0.16	0.2	0.142857	0.186ᘔ35
b−2	0.25	0.3	0.1	0.14	0.0625	0.09	0.04	0.05915343ᘔ0 Ɛ62ᘔ68781Ɛ	0.027	0.04	0.0204081632653 06122448979591 836734693877551	0.02Ɛ322547ᘔ05ᘔ 644ᘔ9380Ɛ908996 741Ɛ615771283Ɛ

Exponent	b=8		b=9		b=10		b=11		b=12
	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.
b6	262,144	107,854	531,441	217,669	1,000,000	402,854	1,771,561	715,261	2,985,984	1,000,000
b5	32,768	16,Ɛ68	59,049	2ᘔ,209	100,000	49,ᘔ54	161,051	79,24Ɛ	248,832	100,000
b4	4,096	2,454	6,561	3,969	10,000	5,954	14,641	8,581	20,736	10,000
b3	512	368	729	509	1,000	6Ɛ4	1,331	92Ɛ	1,728	1,000
b2	64	54	81	69	100	84	121	ᘔ1	144	100
b1	8	8	9	9	10	ᘔ	11	Ɛ	12	10
b−1	0.125	0.16	0.1	0.14	0.1	0.12497	0.09	0.1	0.083	0.1
b−2	0.015625	0.023	0.012345679	0.0194	0.01	0.015343ᘔ0Ɛ6 2ᘔ68781Ɛ059	0.00826446280 99173553719	0.0123456789Ɛ	0.00694	0.01

Some properties

(In this section, all numbers are written with duodecimal, using ᘔ for ten and Ɛ for eleven, unless other bases mentioned)

All squares end with square digits (i.e. end with 0, 1, 4 or 9), if n is divisible by both 2 and 3, then n² ends with 0, if n is not divisible by 2 or 3, then n² ends with 1, if n is divisible by 2 but not by 3, then n² ends with 4, if n is not divisible by 2 but by 3, then n² ends with 9. If the unit digit of n² is 0, then the dozens digit of n² is either 0 or 3, if the unit digit of n² is 1, then the dozens digit of n² is even, if the unit digit of n² is 4, then the dozen digit of n² is 0, 1, 4, 5, 8 or 9, if the unit digit of n² is 9, then the dozen digit of n² is either 0 or 6. (More specially, all squares of (primes ≥ 5) end with 1)

The digital root of a square is 1, 3, 4, 5 or Ɛ.

No repdigits with more than one digit are squares, in fact, a square cannot end with three same digits except 000. (In contrast, in the decimal (base ᘔ) system squares may end in 444, the smallest example is 32² = ᘔ04 = 1444_(ᘔ))

No four-digit palindromic numbers are squares. (we can easily to prove it, since all four-digit palindromic number are divisible by 11, and since they are squares, thus they must be divisible by 11² = 121, and the only four-digit palindromic number divisible by 121 are 1331, 2662, 3993, 5225, 6556, 7887, 8ƐƐ8, 9119, ᘔ44ᘔ and Ɛ77Ɛ, but none of them are squares)

n	n-digit palindromic squares	square roots	number of n-digit palindromic squares
1	1, 4, 9	1, 2, 3	3
2	none	none	0
3	121, 484	11, 22	2
4	none	none	0
5	10201, 12321, 14641, 16661, 16Ɛ61, 40804, 41414, 44944	101, 111, 121, 12Ɛ, 131, 202, 204, 212	8
6	160061	42Ɛ	1
7	1002001, 102ᘔ201, 1093901, 1234321, 148ᘔ841, 4008004, 445ᘔ544, 49ᘔᘔᘔ94	1001, 1015, 1047, 1111, 1221, 2002, 2112, 2244	8
8	none	none	0
9	100020001, 102030201, 104060401, 1060Ɛ0601, 121242121, 123454321, 125686521, 1420Ɛ0241, 1444ᘔ4441, 1468Ɛ8641, 14ᘔ797ᘔ41, 1621Ɛ1261, 163151361, 1ᘔᘔ222ᘔᘔ1, 400080004, 404090404, 410212014, 4414ᘔ4144, 4456Ɛ6544, 496787694, 963848369	10001, 10101, 10201, 10301, 11011, 11111, 11211, 11Ɛ21, 12021, 12121, 1229Ɛ, 1292Ɛ, 12977, 14685, 20002, 20102, 20304, 21012, 21112, 22344, 31053	19
ᘔ	1642662461	434ᘔ5	1
Ɛ	10000200001, 10221412201, 10444ᘔ44401, 12102420121, 12345654321, 141Ɛ1Ɛ1Ɛ141, 14404ᘔ40441, 16497679461, 40000800004, 40441ᘔ14404, 41496869414, 44104ᘔ40144, 49635653694	100001, 101101, 102201, 110011, 111111, 11Ɛ13Ɛ, 120021, 12ᘔ391, 200002, 201102, 204204, 210012, 223344	11
10	none	none	0

It is conjectured that if n is divisible by 4, then there are no n-digit palindromic squares.

R_n² (where R_n is the repunit with length n) is a palindromic number for n ≤ Ɛ, but not for n ≥ 10 (thus, for all odd number n ≤ 19, ther is n-digit palindromic square 123...321), besides, 11ⁿ (also 1{0}1ⁿ, i.e. 101ⁿ, 1001ⁿ, 10001ⁿ, etc.) is a palindromic number for n ≤ 5, but not for n ≥ 6, and it is conjectured that no palindromic numbers are n-th powers if n ≥ 6.

A cube can end with all digits except 2, 6 and ᘔ (in fact, no perfect powers end with 2, 6 or ᘔ), if n is not congruent to 2 mod 4, then n³ ends with the same digit as n; if n is congruent to 2 mod 4, then n³ ends with the digit (the last digit of n +− 6).

The digital root of a cube can be any number.

If k≥2, then n^k+2 ends with the same digit as n^k, thus, if i≥2, j≥2 and i and j have the same parity, then nⁱ and n^j end with the same digit.

Except for 6 and 24, all even perfect numbers end with 54. Additionally, except for 6, 24 and 354, all even perfect numbers end with 054 or 854. Besides, if any odd perfect number exists, then it must end with 1, 09, 39, 69 or 99.

The digital root of an even perfect number is 1, 4, 6 or ᘔ.

Since 10 is the smallest abundant number, all numbers end with 0 are abundant numbers, besides, all numbers end with 6 except 6 itself are also abundant numbers.

unit digit of n^k

k n	0	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10	11	12	13	14	15	16	17	18	19	1ᘔ	1Ɛ	20	Period
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2	1	2	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	2
3	1	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	2
4	1	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	1
5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	2
6	1	6	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	2
8	1	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	2
9	1	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	1
ᘔ	1	ᘔ	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	1
Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	2
10	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
11	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
12	1	2	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	2
13	1	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	3	9	2
14	1	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	1
15	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	5	1	2
16	1	6	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
17	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	7	1	2
18	1	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	8	4	2
19	1	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	1
1ᘔ	1	ᘔ	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	1
1Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	Ɛ	1	2
20	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1

The period of the unit digits of powers of a number must be a divisor of 2 (= λ(10), where λ is the Carmichael function).

n	possible unit digit of an nth power
0	1
1	any number
even number ≥ 2	0, 1, 4, 9 (the square digits)
odd number ≥ 3	0, 1, 3, 4, 5, 7, 8, 9, Ɛ (all digits != 2 mod 4)

final two digits of n^k

k n	0	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10	11	12	13	14	15	16	17	18	19	1ᘔ	1Ɛ	20	Period
0	01	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	1
1	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	1
2	01	02	04	08	14	28	54	ᘔ8	94	68	14	28	54	ᘔ8	94	68	14	28	54	ᘔ8	94	68	14	28	54	6
3	01	03	09	23	69	83	09	23	69	83	09	23	69	83	09	23	69	83	09	23	69	83	09	23	69	4
4	01	04	14	54	94	14	54	94	14	54	94	14	54	94	14	54	94	14	54	94	14	54	94	14	54	3
5	01	05	21	ᘔ5	41	85	61	65	81	45	ᘔ1	25	01	05	21	ᘔ5	41	85	61	65	81	45	ᘔ1	25	01	10
6	01	06	30	60	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	1
7	01	07	41	47	81	87	01	07	41	47	81	87	01	07	41	47	81	87	01	07	41	47	81	87	01	6
8	01	08	54	68	54	68	54	68	54	68	54	68	54	68	54	68	54	68	54	68	54	68	54	68	54	2
9	01	09	69	09	69	09	69	09	69	09	69	09	69	09	69	09	69	09	69	09	69	09	69	09	69	2
ᘔ	01	0ᘔ	84	Ɛ4	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	54	1
Ɛ	01	0Ɛ	ᘔ1	2Ɛ	81	4Ɛ	61	6Ɛ	41	8Ɛ	21	ᘔƐ	01	0Ɛ	ᘔ1	2Ɛ	81	4Ɛ	61	6Ɛ	41	8Ɛ	21	ᘔƐ	01	10
10	01	10	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	1
11	01	11	21	31	41	51	61	71	81	91	ᘔ1	Ɛ1	01	11	21	31	41	51	61	71	81	91	ᘔ1	Ɛ1	01	10
12	01	12	44	08	94	ᘔ8	54	28	14	68	94	ᘔ8	54	28	14	68	94	ᘔ8	54	28	14	68	94	ᘔ8	54	6
13	01	13	69	53	69	53	69	53	69	53	69	53	69	53	69	53	69	53	69	53	69	53	69	53	69	2
14	01	14	94	54	14	94	54	14	94	54	14	94	54	14	94	54	14	94	54	14	94	54	14	94	54	3
15	01	15	01	15	01	15	01	15	01	15	01	15	01	15	01	15	01	15	01	15	01	15	01	15	01	2
16	01	16	30	60	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	1
17	01	17	61	77	01	17	61	77	01	17	61	77	01	17	61	77	01	17	61	77	01	17	61	77	01	4
18	01	18	94	68	14	28	54	ᘔ8	94	68	14	28	54	ᘔ8	94	68	14	28	54	ᘔ8	94	68	14	28	54	6
19	01	19	09	39	69	99	09	39	69	99	09	39	69	99	09	39	69	99	09	39	69	99	09	39	69	4
1ᘔ	01	1ᘔ	44	Ɛ4	94	14	54	94	14	54	94	14	54	94	14	54	94	14	54	94	14	54	94	14	54	3
1Ɛ	01	1Ɛ	81	5Ɛ	41	9Ɛ	01	1Ɛ	81	5Ɛ	41	9Ɛ	01	1Ɛ	81	5Ɛ	41	9Ɛ	01	1Ɛ	81	5Ɛ	41	9Ɛ	01	6
20	01	20	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	1

The period of the final two digits of powers of a number must be a divisor of 10 (= λ(100)).

More generally, for every n≥2, the period of the final n digits of powers of a number must be a divisor of 10ⁿ⁻¹ (= λ(10ⁿ)).

digital root of n^k

k n	0	1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10	11	12	13	14	15	16	17	18	19	1ᘔ	1Ɛ	20	Period
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2	1	2	4	8	5	ᘔ	9	7	3	6	1	2	4	8	5	ᘔ	9	7	3	6	1	2	4	8	5	ᘔ
3	1	3	9	5	4	1	3	9	5	4	1	3	9	5	4	1	3	9	5	4	1	3	9	5	4	5
4	1	4	5	9	3	1	4	5	9	3	1	4	5	9	3	1	4	5	9	3	1	4	5	9	3	5
5	1	5	3	4	9	1	5	3	4	9	1	5	3	4	9	1	5	3	4	9	1	5	3	4	9	5
6	1	6	3	7	9	ᘔ	5	8	4	2	1	6	3	7	9	ᘔ	5	8	4	2	1	6	3	7	9	ᘔ
7	1	7	5	2	3	ᘔ	4	6	9	8	1	7	5	2	3	ᘔ	4	6	9	8	1	7	5	2	3	ᘔ
8	1	8	9	6	4	ᘔ	3	2	5	7	1	8	9	6	4	ᘔ	3	2	5	7	1	8	9	6	4	ᘔ
9	1	9	4	3	5	1	9	4	3	5	1	9	4	3	5	1	9	4	3	5	1	9	4	3	5	5
ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	2
Ɛ	1	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	1
10	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
11	1	2	4	8	5	ᘔ	9	7	3	6	1	2	4	8	5	ᘔ	9	7	3	6	1	2	4	8	5	ᘔ
12	1	3	9	5	4	1	3	9	5	4	1	3	9	5	4	1	3	9	5	4	1	3	9	5	4	5
13	1	4	5	9	3	1	4	5	9	3	1	4	5	9	3	1	4	5	9	3	1	4	5	9	3	5
14	1	5	3	4	9	1	5	3	4	9	1	5	3	4	9	1	5	3	4	9	1	5	3	4	9	5
15	1	6	3	7	9	ᘔ	5	8	4	2	1	6	3	7	9	ᘔ	5	8	4	2	1	6	3	7	9	ᘔ
16	1	7	5	2	3	ᘔ	4	6	9	8	1	7	5	2	3	ᘔ	4	6	9	8	1	7	5	2	3	ᘔ
17	1	8	9	6	4	ᘔ	3	2	5	7	1	8	9	6	4	ᘔ	3	2	5	7	1	8	9	6	4	ᘔ
18	1	9	4	3	5	1	9	4	3	5	1	9	4	3	5	1	9	4	3	5	1	9	4	3	5	5
19	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	ᘔ	1	2
1ᘔ	1	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	Ɛ	1
1Ɛ	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
20	1	2	4	8	5	ᘔ	9	7	3	6	1	2	4	8	5	ᘔ	9	7	3	6	1	2	4	8	5	ᘔ

The period of the digital roots of powers of a number must be a divisor of ᘔ (= λ(Ɛ)).

n	possible digital root of an nth power
0	1
= 1, 3, 7, 9 (mod ᘔ)	any number
= 2, 4, 6, 8 (mod ᘔ)	1, 3, 4, 5, 9, Ɛ
= 5 (mod ᘔ)	1, ᘔ, Ɛ
> 0 and divisible by ᘔ	1, Ɛ

The unit digit of a Fibonacci number can be any digit except 6 (if the unit digit of a Fibonacci number is 0, then the dozens digit of this number must also be 0, thus, all Fibonacci numbers divisible by 6 are also divisible by 100), and the unit digit of a Lucas number cannot be 0 or 9 (thus, no Lucas number is divisible by 10), besides, if a Lucas number ends with 2, then it must end with 0002, i.e., this number is congruent to 2 mod 10⁴.

In the following table, F_n is the n-th Fibonacci number, and L_n is the n-th Lucas number.

n	Fn	digit root of Fn	Ln	digit root of Ln	n	Fn	digit root of Fn	Ln	digit root of Ln
1	1	1	1	1	21	37501	5	81101	Ɛ
2	1	1	3	3	22	5ᘔ301	8	111103	7
3	2	2	4	4	23	95802	2	192204	7
4	3	3	7	7	24	133Ɛ03	ᘔ	2ᘔ3307	3
5	5	5	Ɛ	Ɛ	25	209705	1	47550Ɛ	ᘔ
6	8	8	16	7	26	341608	Ɛ	758816	2
7	11	2	25	7	27	54Ɛ111	1	1012125	1
8	19	ᘔ	3Ɛ	3	28	890719	1	176ᘔ93Ɛ	3
9	2ᘔ	1	64	ᘔ	29	121Ɛ82ᘔ	2	2780ᘔ64	4
ᘔ	47	Ɛ	ᘔ3	2	2ᘔ	1ᘔƐ0347	3	432Ɛ7ᘔ3	7
Ɛ	75	1	147	1	2Ɛ	310ƐƐ75	5	6ᘔƐ0647	Ɛ
10	100	1	22ᘔ	3	30	5000300	8	Ɛ22022ᘔ	7
11	175	2	375	4	31	8110275	2	16110875	7
12	275	3	5ᘔ3	7	32	11110575	ᘔ	25330ᘔᘔ3	3
13	42ᘔ	5	958	Ɛ	33	1922082ᘔ	1	3Ɛ441758	ᘔ
14	6ᘔ3	8	133Ɛ	7	34	2ᘔ3311ᘔ3	Ɛ	6477263Ɛ	2
15	Ɛ11	2	2097	7	35	47551ᘔ11	1	ᘔ3ƐƐ4197	1
16	15Ɛ4	ᘔ	3416	3	36	75882ƐƐ4	1	148766816	3
17	2505	1	54Ɛ1	ᘔ	37	101214ᘔ05	2	23075ᘔ9Ɛ1	4
18	3ᘔƐ9	Ɛ	8907	2	38	176ᘔ979Ɛ9	3	379305607	7
19	6402	1	121Ɛ8	1	39	2780Ɛ0802	5	5ᘔ9ᘔ643Ɛ8	Ɛ
1ᘔ	ᘔ2ƐƐ	1	1ᘔƐ03	3	3ᘔ	432Ɛ885ƐƐ	8	967169ᘔ03	7
1Ɛ	14701	2	310ƐƐ	4	3Ɛ	6ᘔƐ079201	2	13550121ƐƐ	7
20	22ᘔ00	3	50002	7	40	Ɛ22045800	ᘔ	2100180002	3

(Note that F_2ᘔ begins with L_1ᘔ, and F_2Ɛ begins with L_1Ɛ)

The period of the digit root of Fibonacci numbers is ᘔ.

The period of the unit digit of Fibonacci numbers is 20, the final two digits is also 20, the final three digits is 200, the final four digits is 2000, ..., the final n digits is 2×10ⁿ⁻¹ (n ≥ 2).

There are only 13 possible values (of the totally 100 values, thus only 13%) of the final two digits of a Fibonacci number (see (sequence A066853 in the OEIS)).

Except 0 = F₀ and 1 = F₁ = F₂, the only square Fibonacci number is 100 = F₁₀ (100 is the square of 10), thus, 10 is the only base such that 100 is a Fibonacci number (since 100 in a base is just the square of this base, and 0 and 1 cannot be the base of numeral system), and thus we can make the near value of the golden ratio: F₁₁/F₁₀ = 175/100 = 1.75 (since the ratio of two connected Fibonacci numbers is close to the golden ratio, as the numbers get large). Besides, the only cube Fibonacci number is 8 = F₆.

n	2n	n	2n	n	2n	n	2n	n	2n	n	2n
1	2	21	Ɛ2ᘔ20ᘔ8	41	5317Ɛ5804588ᘔ8	61	256906ᘔ1ᘔ93096Ɛ8934ᘔ8	81	11ᘔ12ᘔ743504482569888538ᘔ0ᘔ8	ᘔ1	65933Ɛ8691303ᘔ448Ɛ712227ᘔ7Ɛ11448ᘔ8
2	4	22	1ᘔ584194	42	ᘔ633ᘔƐ408Ɛ5594	62	4Ɛ161183966171Ɛ566994	82	238259286ᘔ08944Ɛ17554ᘔ758194	ᘔ2	10Ɛ667Ɛ51626078895Ɛ22445393ᘔ2289594
3	8	23	38Ɛ48368	43	190679ᘔ815ᘔᘔƐ68	63	9ᘔ302347710323ᘔƐ11768	83	4744Ɛ6551815689ᘔ32ᘔᘔ992Ɛ4368	ᘔ3	21Ɛ113ᘔᘔ305013556Ɛᘔ4488ᘔ76784556Ɛ68
4	14	24	75ᘔ94714	44	361137942Ɛ99Ɛ14	64	1786046932206479ᘔ23314	84	9289Ɛ0ᘔᘔ342Ɛ15786599765ᘔ8714	ᘔ4	43ᘔ2279860ᘔ026ᘔƐ1Ɛ88955931348ᘔƐ1Ɛ14
5	28	25	12Ɛ969228	45	702273685Ɛ77ᘔ28	65	3350091664410937846628	85	16557ᘔ198685ᘔ2Ɛ350Ɛ7730Ɛ95228	ᘔ5	878453750180519ᘔ3Ɛ556ᘔƐ6626959ᘔ3ᘔ28
6	54	26	25Ɛ716454	46	120452714ƐƐ33854	66	66ᘔ0163108821673491054	86	30ᘔƐ3837514Ɛ85ᘔ6ᘔ1Ɛ3261Ɛ6ᘔ454	ᘔ6	15348ᘔ72ᘔ0340ᘔ3787ᘔᘔƐ19Ɛ10516Ɛ787854
7	ᘔ8	27	4ƐƐ2308ᘔ8	47	2408ᘔ5229Ɛᘔ674ᘔ8	67	111803062154431269620ᘔ8	87	619ᘔ7472ᘔ29Ɛ4Ɛ9183ᘔ6503Ɛ188ᘔ8	ᘔ7	2ᘔ695925806818735399ᘔ37ᘔ20ᘔ31Ɛ3534ᘔ8
8	194	28	9Ɛᘔ461594	48	48158ᘔ457Ɛ912994	68	2234061042ᘔ886251704194	88	103792925857ᘔ9Ɛ634790ᘔ07ᘔ35594	ᘔ8	5916Ɛ64Ɛ41143526ᘔ777873841863ᘔ6ᘔ6994
9	368	29	17Ɛ8902Ɛ68	49	942Ɛ588Ɛ3Ɛ625768	69	446810208595504ᘔ3208368	89	20736564Ɛ4Ɛ397Ɛ06936181386ᘔƐ68	ᘔ9	Ɛ631Ɛ09ᘔ82286ᘔ5193335274835079191768
ᘔ	714	2ᘔ	33Ɛ5605Ɛ14	4ᘔ	1685ᘔƐ55ᘔ7Ɛ04Ɛ314	6ᘔ	891420414Ɛ6ᘔᘔ0986414714	8ᘔ	41270Ɛ09ᘔ9ᘔ773ᘔ116703427519Ɛ14	ᘔᘔ	1Ɛ063ᘔ179445518ᘔ36666ᘔ52946ᘔ136363314
Ɛ	1228	2Ɛ	67ᘔƐ00Ɛᘔ28	4Ɛ	314Ɛ9ᘔᘔƐ93ᘔ09ᘔ628	6Ɛ	1562840829Ɛ1981750829228	8Ɛ	82521ᘔ179793278231206852ᘔ37ᘔ28	ᘔƐ	3ᘔ107833688ᘔᘔ358711118ᘔ56918270706628
10	2454	30	1139ᘔ01Ɛ854	50	629Ɛ799Ɛ678179054	70	2Ɛ05481457ᘔ37432ᘔ1456454	90	144ᘔ4383373665344624114ᘔ5873854	Ɛ0	78213467155986Ɛ52222358Ɛ1634521211054
11	48ᘔ8	31	2277803Ɛ4ᘔ8	51	1057Ɛ377Ɛ1343360ᘔ8	71	5ᘔ0ᘔ9428Ɛ3872865828Ɛ08ᘔ8	91	2898874672710ᘔ6890482298Ɛ5274ᘔ8	Ɛ1	1344269122ᘔƐ751ᘔᘔ44446Ɛ5ᘔ3068ᘔ424220ᘔ8
12	9594	32	4533407ᘔ994	52	20Ɛ3ᘔ733ᘔ268670194	72	Ɛ8196855ᘔ752550Ɛ455ᘔ1594	92	557552912522191560944575ᘔᘔ52994	Ɛ2	26885162459Ɛ2ᘔ39888891ᘔƐ86115884844194
13	16Ɛ68	33	8ᘔ668139768	53	41ᘔ792678515120368	73	1Ɛ43714ᘔƐ92ᘔ4ᘔᘔ1ᘔ8ᘔƐ82Ɛ68	93	ᘔƐ2ᘔᘔ5624ᘔ44362Ɛ01688Ɛ2Ɛ98ᘔ5768	Ɛ3	5154ᘔ3048Ɛ7ᘔ58775555639Ɛ5022Ɛ549488368
14	31Ɛ14	34	159114277314	54	839365134ᘔ2ᘔ240714	74	3ᘔ872299Ɛ6589983959Ɛ45Ɛ14	94	19ᘔ598Ɛ049888705ᘔ03155ᘔ5Ɛ758Ɛ314	Ɛ4	ᘔ2ᘔ986095Ɛ38Ɛ532ᘔᘔᘔƐ077ᘔᘔ045ᘔᘔ96954714
15	63ᘔ28	35	2Ɛ6228532628	55	147670ᘔ269858481228	75	79524577Ɛ0Ɛ577476Ɛ7ᘔ8Ɛᘔ28	95	378Ɛ75ᘔ09755520Ɛ8062ᘔƐ8ƐƐ2Ɛ5ᘔ628	Ɛ5	185975016Ɛᘔ75ᘔᘔ65999ᘔ1339808Ɛ99716ᘔ9228
16	107854	36	5Ɛ0454ᘔ65054	56	29312185174Ɛ4942454	76	136ᘔ48Ɛ33ᘔ1ᘔƐ32931Ɛ395Ɛ854	96	735Ɛ2Ɛ8172ᘔᘔᘔ41Ɛ41059Ɛ5Ɛᘔ5ᘔƐ9054	Ɛ6	34Ɛ72ᘔ031Ɛ92Ɛ990Ɛ77782677415Ɛ7723196454
17	2134ᘔ8	37	Ɛᘔ08ᘔ990ᘔ0ᘔ8	57	5662434ᘔ329ᘔ96848ᘔ8	77	271895ᘔ67839ᘔ65663ᘔ76ƐƐ4ᘔ8	97	126Ɛᘔ5Ɛ432599883ᘔ820Ɛ7ᘔƐƐ8Ɛ9Ɛ60ᘔ8	Ɛ7	69Ɛ258063Ɛ65Ɛ761Ɛ3334513282ƐƐ32463708ᘔ8
18	426994	38	1Ɛ81597618194	58	Ɛ104869865797149594	78	52356Ɛ91347790Ɛ107931Ɛᘔ994	98	251Ɛ8Ɛᘔ864Ɛ775479441Ɛ39ƐƐ5Ɛ7Ɛ0194	Ɛ8	117ᘔ4Ɛ4107Ɛ0ƐƐ303ᘔ6668ᘔ26545Ɛᘔ6490721594
19	851768	39	3Ɛ42Ɛ73034368	59	1ᘔ20951750Ɛ372296Ɛ68	79	ᘔ46Ɛ1Ɛ62693361ᘔ213663Ɛ9768	99	4ᘔ3Ɛ5Ɛ9509Ɛ32ᘔ936883ᘔ77ƐᘔƐƐ3ᘔ0368	Ɛ9	23389ᘔ8213ᘔ1Ɛᘔ60791115850ᘔ8ƐƐ90961242Ɛ68
1ᘔ	14ᘔ3314	3ᘔ	7ᘔ85Ɛ26068714	5ᘔ	38416ᘔ32ᘔ1ᘔ724571Ɛ14	7ᘔ	1891ᘔ3Ɛ051667038427107Ɛ7314	9ᘔ	987ᘔƐƐ6ᘔ17ᘔ659671547933Ɛ9Ɛᘔ780714	Ɛᘔ	467579442783Ɛ90136222Ɛ4ᘔ195ƐƐ61702485Ɛ14
1Ɛ	2986628	3Ɛ	1394Ɛᘔ50115228	5Ɛ	74831865839248Ɛ23ᘔ28	7Ɛ	356387ᘔ0ᘔ3112074852213Ɛ2628	9Ɛ	17539ƐƐ183390Ɛ7122ᘔ93667Ɛ7Ɛ9341228	ƐƐ	912Ɛ36885347Ɛ60270445ᘔ9836ƐƐƐ0320494Ɛᘔ28
20	5751054	40	2769Ɛ8ᘔ022ᘔ454	60	12946350Ɛ476495ᘔ47854	80	6Ɛ075381862241294ᘔ4427ᘔ5054	ᘔ0	32ᘔ77Ɛᘔ346761Ɛ2245967113Ɛ3Ɛ6682454	100	1625ᘔ7154ᘔ693Ɛ0052088Ɛ97471ƐƐᘔ0640969Ɛ854

The smallest power of 2 starts with the digit Ɛ is 2²¹ = Ɛ2ᘔ20ᘔ8. For all digits 1 ≤ d ≤ Ɛ, there exists 0 ≤ n ≤ 21 such that 2ⁿ starts with the digit d.

2^1ᘔƐ = 59Ɛ18922Ɛ81631ᘔ39875663Ɛ89ᘔ853ᘔ91Ɛ595336ᘔ6114815ᘔ5ᘔ6929933ᘔ288Ɛ774Ɛ479575ᘔ628 may be the largest power of 2 not contain the digit 0, it has 65 digits. (Note that in the decimal (base ᘔ) system, the largest power of 2 not contain the digit 0 is 2⁷² = Ɛ8196855ᘔ752550Ɛ455ᘔ1594 = 77371252455336267181195264_ᘔ, it has only 22 digits in base ᘔ)

The reciprocal of n is terminating number if and only if n is 3-smooth, the 3-smooth numbers up to 1000 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, 23, 28, 30, 40, 46, 54, 60, 69, 80, 90, ᘔ8, 100, 116, 140, 160, 183, 194, 200, 230, 280, 300, 346, 368, 400, 460, 509, 540, 600, 690, 714, 800, 900, ᘔ16, ᘔ80, 1000.

If and only if n is a divisor of 20, then m² = 1 mod n for every integer m coprime to n.

If and only if n is a divisor of 20, then all Dirichlet characters for n are all real.

If and only if n is a divisor of 20, then n is divisible by all numbers less than or equal to the square root of n.

If and only if n+1 is a divisor of 20, then $C^n_k=\frac{n!}{k!(n-k)!}$ is squarefree for all 0 ≤ k ≤ n.

All prime numbers end with prime digits or 1 (i.e. end with 1, 2, 3, 5, 7 or Ɛ), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or Ɛ (1 and all prime digits that do not divide 10), since all prime numbers other than 2 and 3 are coprime to 10.

The density of primes end with 1 is a relatively low, but the density of primes end with 5, 7 and Ɛ are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or Ɛ)

Except (3, 5), all twin primes end with (5, 7) or (Ɛ, 1).

If n ≥ 3 and n is not divisible by Ɛ, then there are infinitely many primes with digit sum n.

All palindromic primes except 11 has an odd number of digits, since all even-digit palindromic numbers are divisible by 11. The palindromic primes below 1000 are 2, 3, 5, 7, Ɛ, 11, 111, 131, 141, 171, 181, 1Ɛ1, 535, 545, 565, 575, 585, 5Ɛ5, 727, 737, 747, 767, 797, Ɛ1Ɛ, Ɛ2Ɛ, Ɛ6Ɛ.

All lucky numbers end with digit 1, 3, 7 or 9.

Except for 3, all Fermat primes end with 5.

Except for 3, all Mersenne primes end with 7.

Except for 2 and 3, all Sophie Germain primes end with 5 or Ɛ.

Except for 5 and 7, all safe primes end with Ɛ.

A prime p is Gaussian prime (prime in the ring $Z[i]$, where $i=\sqrt{-1}$) if and only if p ends with 7 or Ɛ (or p=3).

A prime p is Eisenstein prime (prime in the ring $Z[\omega]$, where $\omega=\frac{-1+\sqrt{3}i}{2}$) if and only if p ends with 5 or Ɛ (or p=2).

A prime p can be written as x² + y² if and only if p ends with 1 or 5 (or p=2).

A prime p can be written as x² + 3y² if and only if p ends with 1 or 7 (or p=3).

By Midy theorem, if p is a prime with even period length (let its period length be n), then if we let $\frac{a}{p}=0.\overline{a_1a_2a_3...a_n}$, then a_i + a_i+n/2 = Ɛ for every 1 ≤ i ≤ n/2. e.g. 1/5 = 0.249724972497..., and 24 + 97 = ƐƐ, and 1/7 = 0.186ᘔ35186ᘔ35..., and 186 + ᘔ35 = ƐƐƐ, all primes (other than 2 and 3) ≤ 37 except Ɛ, 1Ɛ and 31 have even period length, thus they can use Midy theorem to get an Ɛ-repdigit number, the length of this number is the period length of this prime.

The unique primes below 10⁶⁰ are Ɛ, 11, 111, Ɛ0Ɛ1, ƐƐ01, 11111, 24727225, Ɛ0Ɛ0Ɛ0Ɛ0Ɛ1, Ɛ00Ɛ00ƐƐ0ƐƐ1, 100ƐƐƐᘔƐᘔƐƐ000101, 1111111111111111111, ƐƐƐƐ0000ƐƐƐƐ0000ƐƐƐƐ0001, 100ƐƐƐᘔƐƐ0000ƐƐƐᘔƐƐ000101, 10ƐƐƐᘔᘔᘔƐ011110ƐᘔᘔᘔƐ00011, ƐƐƐƐƐƐƐƐ00000000ƐƐƐƐƐƐƐƐ00000001, ƐƐƐ000000ƐƐƐ000000ƐƐƐƐƐƐ000ƐƐƐƐƐƐ001, and the period length of their reciprocals are 1, 2, 3, ᘔ, 10, 5, 18, 1ᘔ, 19, 50, 17, 48, 70, 5ᘔ, 68, 53.

If p is a safe prime other than 5, 7 and Ɛ, then the period length of 1/p is (p-1)/2. (this is not true for all primes ends with Ɛ (other than Ɛ itself), the first counterexample is p = 2ƐƐ, where the period length of 1/p is only 37)

There is no full reptend prime ends with 1, since 10 is quadratic residue for all primes end with 1. (In contrast, in decimal (base ᘔ) system there are some such primes, and may be infinitely many such primes, the first few such primes in that base are 61_(ᘔ) = 51, 131_(ᘔ) = ᘔƐ, 181_(ᘔ) = 131, 461_(ᘔ) = 325, 491_(ᘔ) = 34Ɛ, see (sequence A073761 in the OEIS)) (if so, then this prime p is a proper prime (i.e. for the reciprocal of such primes (1/p), each digit 0, 1, 2, ..., Ɛ appears in the repeating sequence the same number of times as does each other digit (namely, (p−1)/10 times))) (In fact, not only for base 10 such primes do not exist, for all bases = 0 mod 4 (i.e. bases end with digit 0, 4 or 8), such primes do not exist)

5 and 7 are the only two safe primes which are also full reptend primes, since except 5 and 7, all safe primes end with Ɛ, and 10 is quadratic residue for all primes end with Ɛ. (In contrast, in decimal (base ᘔ) system there may be infinitely many such primes, the first few such primes in that base are 7_(ᘔ) = 7, 23_(ᘔ) = 1Ɛ, 47_(ᘔ) = 3Ɛ, 59_(ᘔ) = 4Ɛ, 167_(ᘔ) = 11Ɛ, see (sequence A000353 in the OEIS)) (if so, then this prime p produces a stream of p−1 pseudo-random digits) (In fact, not only for base 10 there are only finitely many such primes, of course for square bases (bases of the form k²) only 2 may be full reptend prime (if the base is odd), and all odd primes are not full reptend primes, but since all safe primes are odd primes, for these bases such primes do not exist, besides, for the bases of the form 3k², only 5 and 7 can be such primes, the proof for these bases is completely the same as that for base 10)

p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p	p	period length of 1/p
2	0	111	3	267	266	41Ɛ	20Ɛ	591	3ᘔ	767	766	927	926	Ɛ1Ɛ	56Ɛ	1107	1106	12Ɛ5	12Ɛ4	14Ɛ1	14Ɛ	16ᘔ7	16ᘔ6
3	0	117	116	271	27	421	63	59Ɛ	2ᘔƐ	76Ɛ	395	955	954	Ɛ21	570	1115	1114	1301	760	14Ɛ5	14Ɛ4	16Ɛ5	188
5	4	11Ɛ	6Ɛ	277	276	427	426	5Ɛ1	159	771	132	95Ɛ	48Ɛ	Ɛ25	Ɛ24	1125	1124	1317	506	14ƐƐ	85Ɛ	16Ɛ7	16Ɛ6
7	6	125	124	27Ɛ	13Ɛ	431	109	5Ɛ5	5Ɛ4	775	774	965	964	Ɛ2Ɛ	575	112Ɛ	675	1337	512	150Ɛ	865	1705	398
Ɛ	1	12Ɛ	75	285	284	435	434	5Ɛ7	66	77Ɛ	39Ɛ	971	172	Ɛ31	116	1135	1134	133Ɛ	77Ɛ	1517	1ᘔᘔ	1711	493
11	2	131	76	291	83	437	436	5ƐƐ	2ƐƐ	785	784	987	32ᘔ	Ɛ37	Ɛ36	114Ɛ	685	1345	1344	1521	436	1715	1714
15	14	13Ɛ	7Ɛ	295	294	447	446	611	163	791	3ᘔ6	995	994	Ɛ45	Ɛ44	1151	115	1351	166	1525	1524	1727	1726
17	6	141	20	2ᘔ1	150	455	454	615	614	797	796	9ᘔ7	9ᘔ6	Ɛ61	16	1165	1164	1365	1364	1547	1Ɛ2	1735	1734
1Ɛ	Ɛ	145	144	2ᘔƐ	155	457	15ᘔ	617	206	7ᘔ1	138	9ᘔƐ	4Ɛ5	Ɛ67	3ᘔ2	1167	42	1367	1366	1561	89	1745	1744
25	4	147	56	2Ɛ1	26	45Ɛ	22Ɛ	61Ɛ	30Ɛ	7ƐƐ	3ƐƐ	9Ɛ1	9Ɛ	Ɛ6Ɛ	595	1185	1184	136Ɛ	795	156Ɛ	97	1747	1746
27	26	157	12	2ƐƐ	37	465	464	637	212	801	140	9Ɛ5	9Ɛ4	Ɛ71	596	118Ɛ	6ᘔ5	1377	106	1577	1576	1751	32ᘔ
31	9	167	166	301	90	46Ɛ	7	63Ɛ	31Ɛ	80Ɛ	405	9ƐƐ	4ƐƐ	Ɛ91	2Ɛ3	1197	472	138Ɛ	7ᘔ5	157Ɛ	89Ɛ	1755	1754
35	34	16Ɛ	95	307	102	471	13	647	216	817	816	ᘔ07	ᘔ06	Ɛ95	Ɛ94	11ᘔ1	6Ɛ	1391	3Ɛ3	1585	1584	1757	1756
37	36	171	96	30Ɛ	165	481	24	655	654	825	824	ᘔ0Ɛ	505	Ɛ97	Ɛ96	11ᘔ5	11ᘔ4	1395	1394	1587	2ᘔ	176Ɛ	995
3Ɛ	1Ɛ	175	8	315	314	485	44	661	176	82Ɛ	415	ᘔ11	56	Ɛᘔ5	Ɛᘔ4	11ᘔ7	11ᘔ6	13ᘔ1	7Ɛ0	1591	2Ɛ6	1781	4Ɛ0
45	44	17Ɛ	9Ɛ	321	170	48Ɛ	245	665	138	835	834	ᘔ17	ᘔ16	ƐƐ5	ƐƐ4	11ᘔƐ	6Ɛ5	13ᘔ7	536	15ᘔƐ	8Ɛ5	1785	1784
4Ɛ	25	181	ᘔ0	325	324	497	496	66Ɛ	335	841	84	ᘔ27	156	ƐƐ7	ƐƐ6	11Ɛ7	11Ɛ6	13Ɛ1	13Ɛ	15ƐƐ	8ƐƐ	178Ɛ	9ᘔ5
51	13	18Ɛ	ᘔ5	327	10ᘔ	4ᘔ5	4ᘔ4	675	674	851	14ᘔ	ᘔ35	ᘔ34	1005	1004	1201	700	13Ɛ5	13Ɛ4	1601	160	1797	1796
57	56	195	194	32Ɛ	175	4Ɛ1	9ᘔ	687	686	855	854	ᘔ37	ᘔ36	1011	73	120Ɛ	705	1405	1404	1615	1614	17ᘔ1	9Ɛ0
5Ɛ	2Ɛ	19Ɛ	ᘔƐ	33Ɛ	17Ɛ	4ƐƐ	25Ɛ	68Ɛ	345	85Ɛ	42Ɛ	ᘔ3Ɛ	51Ɛ	1017	1016	1211	706	1407	326	1621	910	17ᘔ5	17ᘔ4
61	30	1ᘔ5	1ᘔ4	347	46	507	182	695	694	865	864	ᘔ41	188	1021	610	121Ɛ	70Ɛ	1425	1424	1625	1624	17ƐƐ	9ƐƐ
67	22	1ᘔ7	46	34Ɛ	2Ɛ	511	266	69Ɛ	34Ɛ	867	2ᘔ2	ᘔ45	ᘔ44	1027	1026	1231	123	142Ɛ	815	1635	274	1807	682
6Ɛ	35	1Ɛ1	Ɛ6	357	11ᘔ	517	516	6ᘔ7	6ᘔ6	871	152	ᘔ4Ɛ	525	1041	620	123Ɛ	71Ɛ	1431	286	1647	276	1815	1814
75	8	1Ɛ5	1Ɛ4	35Ɛ	18Ɛ	51Ɛ	45	6Ɛ1	6Ɛ	881	440	ᘔ5Ɛ	52Ɛ	1047	1046	1245	1244	1437	1436	1655	1654	181Ɛ	ᘔ0Ɛ
81	14	1Ɛ7	1Ɛ6	365	364	527	526	701	360	88Ɛ	445	ᘔ6Ɛ	535	104Ɛ	625	1255	114	143Ɛ	81Ɛ	1657	61ᘔ	1825	1824
85	84	205	204	375	34	531	276	705	704	8ᘔ5	98	ᘔ77	ᘔ76	1051	313	1257	49ᘔ	1445	1444	165Ɛ	92Ɛ	1831	509
87	86	217	86	377	376	535	534	70Ɛ	365	8ᘔ7	2Ɛ6	ᘔ87	ᘔ86	1061	16	125Ɛ	72Ɛ	1457	1456	1667	622	183Ɛ	ᘔ1Ɛ
8Ɛ	45	21Ɛ	10Ɛ	391	1ᘔ6	541	54	711	71	8ᘔƐ	455	ᘔ91	283	106Ɛ	635	1261	730	1461	38	1671	936	184Ɛ	ᘔ25
91	46	221	66	397	396	545	544	71Ɛ	36Ɛ	8Ɛ5	8Ɛ4	ᘔ95	ᘔ94	107Ɛ	63Ɛ	126Ɛ	735	1465	1464	1677	20ᘔ	1861	269
95	94	225	224	3ᘔ5	3ᘔ4	557	556	721	370	8Ɛ7	8Ɛ6	ᘔ9Ɛ	54Ɛ	1087	1086	127Ɛ	73Ɛ	1467	562	167Ɛ	93Ɛ	1865	1864
ᘔ7	ᘔ6	237	92	3ᘔƐ	1Ɛ5	565	564	727	24ᘔ	901	230	ᘔᘔ7	376	109Ɛ	64Ɛ	1281	740	1471	419	1681	140	186Ɛ	ᘔ35
ᘔƐ	55	241	120	3Ɛ5	3Ɛ4	575	574	735	734	905	198	ᘔᘔƐ	555	10Ɛ1	329	1295	94	1475	1474	1685	1684	1875	1874
Ɛ5	Ɛ4	24Ɛ	125	3Ɛ7	3Ɛ6	577	116	737	736	907	906	ᘔƐ7	ᘔƐ6	10Ɛ7	10Ɛ6	1297	1296	147Ɛ	83Ɛ	168Ɛ	945	1877	146
Ɛ7	Ɛ6	251	73	401	100	585	584	745	744	90Ɛ	465	ᘔƐƐ	55Ɛ	10ƐƐ	65Ɛ	12ᘔ1	75	148Ɛ	845	1697	1696	189Ɛ	ᘔ4Ɛ
105	104	255	254	40Ɛ	205	587	1ᘔᘔ	747	9ᘔ	91Ɛ	46Ɛ	Ɛ11	1ᘔ2	1101	220	12ᘔ5	2Ɛ8	1495	1494	169Ɛ	94Ɛ	18ᘔ1	210
107	106	25Ɛ	12Ɛ	415	414	58Ɛ	2ᘔ5	751	1ᘔ3	921	236	Ɛ15	228	1105	1104	12ᘔ7	12ᘔ6	149Ɛ	84Ɛ	16ᘔ1	950	18ᘔƐ	ᘔ55

period length	primes	period length	primes
1	Ɛ	11	1Ɛ0411, 69ᘔ3901
2	11	12	157, 7687
3	111	13	51, 471, 57Ɛ1
4	5, 25	14	15, 81, 106ᘔ95
5	11111	15	ᘔ9ᘔ9ᘔƐ, 126180ƐƐ0ƐƐ
6	7, 17	16	Ɛ61, 1061
7	46Ɛ, 2ᘔ3Ɛ	17	1111111111111111111
8	75, 175	18	24727225
9	31, 3ᘔ891	19	Ɛ00Ɛ00ƐƐ0ƐƐ1
ᘔ	Ɛ0Ɛ1	1ᘔ	Ɛ0Ɛ0Ɛ0Ɛ0Ɛ1
Ɛ	1Ɛ, 754Ɛ2Ɛ41	1Ɛ	3Ɛ, 78935Ɛᘔ441, 523074ᘔ3ᘔᘔƐ
10	ƐƐ01	20	141, 8Ɛ5281

The period level of a prime p ≥ 5 is (p−1)/(period length of 1/p), e.g., $\frac{1}{17}$ has period level 3, thus the numbers $\frac{a}{17}$ with integer 1 ≤ a ≤ 16 from 3 different cycles: 076Ɛ45 (for a = 1, 7, 8, Ɛ, 10, 16), 131ᘔ8ᘔ (for a = 2, 3, 5, 12, 14, 15) and 263958 (for a = 4, 6, 9, ᘔ, 11, 13). Besides, $\frac{1}{15}$ has period level 1, thus this number is a cyclic number and 15 is a full-reptend prime, and all of the numbers $\frac{a}{15}$ with integer 1 ≤ a ≤ 14 from the cycle 08579214Ɛ36429ᘔ7.

smallest multiple of n uses only 0's and 1's

n	+1	+2	+3	+4	+5	+6	+7	+8	+9	+ᘔ	+Ɛ	+10
0+	1	10	10	10	101	10	1001	100	100	1010	11111111111	10
10+	11	10010	1010	100	10111	100	1001	1010	10010	111111111110	11101	100
20+	110111	110	1000	10010	101	1010	101011	1000	111111111110	101110	101101	100
30+	1001001	10010	110	10100	111001	10010	101001	111111111110	10100	111010	10001111	100
40+	10111101	1101110	101110	110	1100101	1000	1011101111111	100100	10010	1010	101011	1010
50+	10010101	1010110	100100	1000	1111	111111111110	1100101	101110	111010	1011010	1100111	100
60+	10101101	10010010	1101110	10010	1011101111111	110	10101011	10100	10000	1110010	100111001	10010
70+	1101001	1010010	1010	1111111111100	10001	10100	1001	111010	1010110	100011110	101101	1000
80+	111011	101111010	1111111111100	1101110	110001	101110	10100111	1100	1011010	11001010	100111	1000
90+	1010111111	10111011111110	10010010	100100	10101001	10010	110101001	1010	1100	1010110	101100011	10100
ᘔ0+	111111111101	100101010	1110010	1010110	11100001	100100	1100001	10000	1010010	11110	1111011111	111111111110
Ɛ0+	1001	11001010	101000	1011100	101011	111010	11010111	1011010	100011110	11001110	1111111111111111111111	100

n	1	5	7	Ɛ	11	15	17	1Ɛ	21	25	27	2Ɛ
smallest k such that k×n is a repunit	1	275	1ᘔ537	123456789Ɛ	1	92ᘔ79Ɛ43715865	8327	69Ɛ63848Ɛ	634ᘔ159788253ᘔ72Ɛ1	55	509867481Ɛ793ᘔᘔ5ᘔ1243628Ɛ317	45ᘔ3976ᘔ7Ɛ
the length of the repunit k×n	1	4	6	Ɛ	2	14	6	Ɛ	18	4	26	10

(this k is usually not prime, in fact, this k is not prime for all numbers n < 100 which are coprime to 10 except n = 55, and for n < 1000 which is coprime to 10, this k is prime only for n = 55, 101, 19Ɛ, 275 and 46Ɛ, and only 19Ɛ and 46Ɛ are itself prime, other 3 numbers are 5×11, 5×25 and 11×25, and this k for these n are successively 25, 11 and 5, which makes k×n = R₄ = 1111 = 5×11×25, besides, this k for n = 46Ɛ is 2ᘔ3Ɛ, which makes k×n = R₇ = 1111111, a repunit semiprime, and this k for n = 19Ɛ is a ᘔ8-digit prime number, with k×n = R_ᘔƐ, another repunit semiprime)

For every prime p except Ɛ, the repunit with length p is congruent to 1 mod p. (The converse is also not true, the counterexamples up to 1000 are 4, 6, 10, 33, 55, 77, Ɛ1, 101, 187, 1Ɛ0, 275, 444, 4ᘔ7, 777, 781, Ɛ55, they are called "repunit pseudoprimes" (or weak deceptive primes), all deceptive primes are also repunit pseudoprimes, if n is repunit pseudoprime, then R_n is also repunit pseudoprime, thus there are infinitely may repunit pseudoprimes. No repunit pseudoprimes are divisible by 8, 9 or Ɛ. (in fact, the repunit pseudoprimes are exactly the weak pseudoprimes for base 10 (composite numbers c such that 10^c = 10 mod c) which are not divisible by Ɛ) Besides, the deceptive primes are exactly the repunit pseudoprimes which are coprime to 10)

Smallest multiple of n with digit sum 2 are: (0 if not exist)

2, 2, 20, 20, 101, 20, 1001, 20, 200, 1010, 0, 20, 11, 10010, 1010, 200, 100000001, 200, 1001, 1010, 10010, 0, 0, 20, 10000000001, 110, 2000, 10010, 101, 1010, 1000000000000001, 200, 0, 1000000010, 1000000000001, 200, ..., if and only if n is divisible by some prime p with 1/p odd period length, then such number does not exist.

Smallest multiple of n with digit sum 3 are: (0 if not exist)

3, 12, 3, 30, 21, 30, 12, 120, 30, 210, 0, 30, 0, 12, 210, 300, 201, 30, 10101, 210, 120, 0, 1010001, 120, 21, 0, 300, 120, 0, 210, 1010001, 1200, 0, 2010, 200001, 30, ..., such number does not exist for n divisible by Ɛ, 11 or 25.

Smallest multiple of n with digit sum 4 are: (0 if not exist)

4, 4, 13, 4, 13, 40, 103, 40, 130, 130, 0, 40, 22, 1030, 13, 40, 3001, 130, 2002, 130, 103, 0, 11101, 40, 10012, 22, 1300, 1030, 202, 130, 10003, 400, 0, 30010, 101101, 130, ..., such number is conjectured to exist for all n not divisible by Ɛ (of course, if n is divisible by Ɛ, then such number does not exist).

Smallest multiple of n with digit sum n are:

1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 1Ɛ0, 20Ɛ, 22ᘔ, 249, 268, 287, 2ᘔ6, 45ᘔ, 488, 4Ɛ6, 1Ɛᘔ, 8Ɛ4, 3Ɛᘔ0, 3ƐƐ, 23Ɛᘔ, 1899, ᘔᘔ8, 2Ɛ79, 4Ɛ96, 1Ɛᘔ9, 4ᘔᘔ8, 2ƐƐ9, 3ᘔƐᘔ, 799ᘔ, 5ƐƐ90, ..., such number is conjectured to exist for all n.

All numbers of the form 34{1} are composite (proof: 34{1_n} = 34×10ⁿ+(10ⁿ−1)/Ɛ = (309×10ⁿ−1)/Ɛ and it can be factored to ((19×10^n/2−1)/Ɛ) × (19×10^n/2+1) for even n and divisible by 11 for odd n). Besides, 34 was proven to be the smallest n such that all numbers of the form n{1} are composite. However, the smallest prime of the form 23{1} is 23{1_Ɛ78}, it has Ɛ7ᘔ digits. The only other two n≤100 such that all numbers of the form n{1} are composite are 89 and 99 (the reason of 89 is the same as 34, and the reason of 99 is 99{1_n} is divisible by 5, 11 or 25).

The only known of the form 1{0}1 is 11, these are the primes obtained as the concatenation of a power of 10 followed by a 1. If n = 1 mod 11, then all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are divisible by 11 and thus composite. Except 10, the smallest n not = 1 mod 11 such that all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are composite was proven by Ɛᘔ, since all numbers obtained as the concatenation of a power of Ɛᘔ (>1) followed by a 1 are divisible by either Ɛ or 11 and thus composite. However, the smallest prime obtained as the concatenation of a power of 58 (>1) followed by a 1 is 10×58^2781Ɛ5+1, it has 459655 digits.

All numbers of the form 1{5}1 are composite (proof: 1{5_n}1 = (14×10ⁿ⁺¹−41)/Ɛ and it can be factored to (4×10^(n+1)/2−7) × ((4×10^(n+1)/2−7)/Ɛ) for odd n and divisible by 11 for even n).

The emirps below 1000 are 15, 51, 57, 5Ɛ, 75, Ɛ5, 107, 117, 11Ɛ, 12Ɛ, 13Ɛ, 145, 157, 16Ɛ, 17Ɛ, 195, 19Ɛ, 1ᘔ7, 1Ɛ5, 507, 51Ɛ, 541, 577, 587, 591, 59Ɛ, 5Ɛ1, 5ƐƐ, 701, 705, 711, 751, 76Ɛ, 775, 785, 7ᘔ1, 7ƐƐ, Ɛ11, Ɛ15, Ɛ21, Ɛ31, Ɛ61, Ɛ67, Ɛ71, Ɛ91, Ɛ95, ƐƐ5, ƐƐ7.

The non-repdigit permutable primes below 10¹⁰¹⁰⁰ are 15, 57, 5Ɛ, 117, 11Ɛ, 5ƐƐƐ (the smallest representative prime of the permutation set).

The non-repdigit circular primes below 10¹⁰¹⁰⁰ are 15, 57, 5Ɛ, 117, 11Ɛ, 175, 1Ɛ7, 157Ɛ, 555Ɛ, 115Ɛ77 (the smallest representative prime of the cycle).

The first few Smarandache primes are the concatenation of the first 5, 15, 4Ɛ, 151, ... positive integers.

The only known Smarandache–Wellin primes are 2 and 2357Ɛ11.

There are exactly 15 minimal primes, and they are 2, 3, 5, 7, Ɛ, 11, 61, 81, 91, 401, ᘔ41, 4441, ᘔ0ᘔ1, ᘔᘔᘔᘔ1, 44ᘔᘔᘔ1, ᘔᘔᘔ0001, ᘔᘔ000001.

The smallest weakly prime is 6Ɛ8ᘔƐ77.

The largest left-truncatable prime is 28-digit 471ᘔ34ᘔ164259Ɛᘔ16Ɛ324ᘔƐ8ᘔ32Ɛ7817, and the largest right-truncatable prime is ᘔ-digit 375ƐƐ5Ɛ515.

The only two base 10 Wieferich primes up to 10¹⁰ are 1685 and 5Ɛ685, note that both of the numbers end with 685, and it is conjectured that all base 10 Wieferich primes end with 685. (there is also a note for the only two known base 2 Wieferich primes (771 and 2047) minus 1 written in base 2, 8 (= 2³) and 14 (= 2⁴), 770 = 010001000100₍₂₎ = 444₍₁₄₎ is a repdigit in base 14, and 2046 = 110110110110₍₂₎ = 6666₍₈₎ is also a repdigit in base 8, see Wieferich prime#Binary periodicity of p − 1)

There are 1, 2, 3, 5 and 6-digit (but not 4-digit) narcissistic numbers, there are totally 73 narcissistic numbers, the first few of which are 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 25, ᘔ5, 577, 668, ᘔ83, 14765, 938ᘔ4, 369862, ᘔ2394ᘔ, ..., the largest of which is 43-digit 15079346ᘔ6Ɛ3Ɛ14ƐƐ56Ɛ395898Ɛ96629ᘔ8Ɛ01515344Ɛ4Ɛ0714Ɛ. (see (sequence A161949 in the OEIS))

The only two factorions are 1 and 2.

The only seven happy numbers below 1000 are 1, 10, 100, 222, 488, 848 and 884, almost all natural numbers are unhappy. All unhappy numbers get to one of these four cycles: {5, 21}, {8, 54, 35, 2ᘔ, 88, ᘔ8, 118, 56, 51, 22}, {18, 55, 42}, {68, 84}, or one of the only two fixed points other than 1: 25 and ᘔ5. (In contrast, in the decimal (base ᘔ) system there are Ɛᘔ happy numbers below 1000_(ᘔ) (=6Ɛ4), and all unhappy number get to this cycle: {4, 16, 37, 58, 89, 145, 42, 20}, there are no fixed points other than 1)

If we use the sum of the cubes (instead of squares) of the digits, then every natural numbers get to either 1 or the cycle {8, 368, 52Ɛ, ᘔ20, 700, 247, 2ᘔ7, 947, 7ᘔ8, 10ᘔ7, 940, 561, 246, 200}. (In contrast, in the decimal (base ᘔ) system all multiple of 3 get to 153_(ᘔ) (=109), and other numbers get to either one of these four fixed points: 1, 370, 371, 407, or one of these four cycles: {55, 250, 133}, {136, 244}, {160, 217, 352}, {919, 1459}) (for the example of the famous Hardy–Ramanujan number 1001 = 9³ + ᘔ³, we know that this sequence with initial term 9ᘔ is 9ᘔ, 1001, 2, 8, 368, 52Ɛ, ᘔ20, 700, 247, 2ᘔ7, 947, 7ᘔ8, 10ᘔ7, 940, 561, 246, 200, 8, 368, 52Ɛ, ᘔ20, 700, 247, 2ᘔ7, 947, 7ᘔ8, 10ᘔ7, 940, 561, 246, 200, 8, ...)

n	fixed points and cycled for the sequence for sum of n-th powers of the digits	length of these cycles
1	{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {ᘔ}, {Ɛ}	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2	{1}, {5, 21}, {8, 54, 35, 2ᘔ, 88, ᘔ8, 118, 56, 51, 22}, {18, 55, 42}, {25}, {68, 84}, {ᘔ5}	1, 2, ᘔ, 3, 1, 2, 1
3	{1}, {8, 368, 52Ɛ, ᘔ20, 700, 247, 2ᘔ7, 947, 7ᘔ8, 10ᘔ7, 940, 561, 246, 200}, {577}, {668}, {6Ɛ5, Ɛ74, 100ᘔ}, {ᘔ83}, {11ᘔᘔ}	1, 12, 1, 1, 3, 1, 1
4	{1}, {ᘔ6ᘔ, 103ᘔ8, 8256, 35ᘔ9, 9ƐᘔƐ, 22643, Ɛ69, 1102ᘔ, 596ᘔ, ᘔ842, 8394, 6442, 1080, 2455}, {206ᘔ, 6668, 4754}, {3ᘔ2Ɛ, 12396, 472Ɛ, ᘔ02ᘔ, Ɛ700, 9ᘔ42, 98ᘔ9, 13902}	1, 12, 3, 8

The harshad numbers up to 200 are 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 10, 1ᘔ, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, ᘔ0, ᘔ1, Ɛ0, 100, 10ᘔ, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1ᘔ0, 1Ɛ0, 1Ɛᘔ, 200, although the sequence of factorials begins with harshad numbers, not all factorials are harshad numbers, after 7! (=2Ɛ00, with digit sum 11 but 11 does not divide 7!), 8ᘔ4! is the next that is not (8ᘔ4! has digit sum 8275 = Ɛ×8Ɛ7, thus not divide 8ᘔ4!). There are no 21 consecutive integers that are all harshad numbers, but there are infinitely many 20-tuples of consecutive integers that are all harshad numbers.

The Kaprekar numbers up to 10000 are 1, Ɛ, 56, 66, ƐƐ, 444, 778, ƐƐƐ, 12ᘔᘔ, 1640, 2046, 2929, 3333, 4973, 5Ɛ60, 6060, 7249, 8889, 9293, 9Ɛ76, ᘔ580, ᘔ912, ƐƐƐƐ.

The Kaprekar's routine of any four-digit number which is not repdigit converges to either the cycle {3ƐƐ8, 8284, 6376} or the cycle {4198, 8374, 5287, 6196, 7ƐƐ4, 7375}, and the Kaprekar map of any three-digit number which is not repdigit converges to the fixed point 5Ɛ6, and the Kaprekar map of any two-digit number which is not repdigit converges to the cycle {0Ɛ, ᘔ1, 83, 47, 29, 65}.

n	cycles for Kaprekar's routine for n-digit numbers	length of these cycles
1	{0}	1
2	{00}, {0Ɛ, ᘔ1, 83, 47, 29, 65}	1, 6
3	{000}, {5Ɛ6}	1, 1
4	{0000}, {3ƐƐ8, 8284, 6376}, {4198, 8374, 5287, 6196, 7ƐƐ4, 7375}	1, 3, 6
5	{00000}, {64Ɛ66, 6ƐƐƐ5}, {83Ɛ74}	1, 2, 1
6	{000000}, {420ᘔ98, ᘔ73742, 842874, 642876, 62ƐƐ86, 951963, 860ᘔ54, ᘔ40ᘔ72, ᘔ82832, 864654}, {65ƐƐ56}	1, ᘔ, 1

The self numbers up to 600 are 1, 3, 5, 7, 9, Ɛ, 20, 31, 42, 53, 64, 75, 86, 97, ᘔ8, Ɛ9, 10ᘔ, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1ᘔ9, 1Ɛᘔ, 20Ɛ, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2ᘔᘔ, 2ƐƐ, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39ᘔ, 3ᘔƐ, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48ᘔ, 49Ɛ, 4Ɛ0, 501, 512, 514, 525, 536, 547, 558, 569, 57ᘔ, 58Ɛ, 5ᘔ0, 5Ɛ1.

The Friedman numbers up to 1000 are 121=11², 127=7×21, 135=5×31, 144=4×41, 163=3×61, 368=8⁶⁻³, 376=6×73, 441=(4+1)⁴, 445=5⁴+4.

The Keith numbers up to 1000 are 11, 15, 1Ɛ, 22, 2ᘔ, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, ᘔᘔ, ƐƐ, 125, 215, 24ᘔ, 405, 42ᘔ, 654, 80ᘔ, 8ᘔ3, ᘔ59.

There are totally 71822 polydivisible numbers, the largest of which is 24-digit 606890346850Ɛᘔ6800Ɛ036206464. However, there are no Ɛ-digit polydivisible numbers contain the digits 1 to Ɛ exactly once each.

The candidate Lychrel numbers up to 1000 are 179, 1Ɛ9, 278, 2Ɛ8, 377, 3Ɛ7, 476, 4Ɛ6, 575, 5Ɛ5, 674, 6Ɛ4, 773, 7Ɛ3, 872, 8Ɛ2, 971, 9Ɛ1, ᘔ2Ɛ, ᘔ3Ɛ, ᘔ5Ɛ, ᘔ70, ᘔᘔƐ, ᘔƐ0, Ɛ2ᘔ, Ɛ3ᘔ, Ɛ5ᘔ, Ɛᘔᘔ. The only suspected Lychrel seed numbers up to 1000 are 179, 1Ɛ9, ᘔ3Ɛ and ᘔ5Ɛ. However, it is unknown whether any Lychrel number exists. (Lychrel numbers only known to exist in these bases: Ɛ, 15, 18, 22 and all powers of 2)

Most numbers that end with 2 are nontotient (in fact, all nontotients < 58 except 2ᘔ end with 2), except 2 itself, the first counterexample is 92, which equals φ(ᘔ1) = φ(Ɛ²) and φ(182) = φ(2×Ɛ²), next counterexample is 362, which equals φ(381) = φ(1Ɛ²) and φ(742) = φ(2×1Ɛ²), there are only 9 such numbers ≤ 10000 (the number 2 itself is not counted), all such numbers (except the number 2 itself) are of the form φ(p²) = p(p−1), where p is a prime ends with Ɛ.

Prime numbers

(In this section, all numbers are written with duodecimal)

A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself. Natural numbers greater than 1 that are not prime are called composite.

The first 1ᘔ5 prime numbers (all the prime numbers less than 1000) are:

2, 3, 5, 7, Ɛ, 11, 15, 17, 1Ɛ, 25, 27, 31, 35, 37, 3Ɛ, 45, 4Ɛ, 51, 57, 5Ɛ, 61, 67, 6Ɛ, 75, 81, 85, 87, 8Ɛ, 91, 95, ᘔ7, ᘔƐ, Ɛ5, Ɛ7, 105, 107, 111, 117, 11Ɛ, 125, 12Ɛ, 131, 13Ɛ, 141, 145, 147, 157, 167, 16Ɛ, 171, 175, 17Ɛ, 181, 18Ɛ, 195, 19Ɛ, 1ᘔ5, 1ᘔ7, 1Ɛ1, 1Ɛ5, 1Ɛ7, 205, 217, 21Ɛ, 221, 225, 237, 241, 24Ɛ, 251, 255, 25Ɛ, 267, 271, 277, 27Ɛ, 285, 291, 295, 2ᘔ1, 2ᘔƐ, 2Ɛ1, 2ƐƐ, 301, 307, 30Ɛ, 315, 321, 325, 327, 32Ɛ, 33Ɛ, 347, 34Ɛ, 357, 35Ɛ, 365, 375, 377, 391, 397, 3ᘔ5, 3ᘔƐ, 3Ɛ5, 3Ɛ7, 401, 40Ɛ, 415, 41Ɛ, 421, 427, 431, 435, 437, 447, 455, 457, 45Ɛ, 465, 46Ɛ, 471, 481, 485, 48Ɛ, 497, 4ᘔ5, 4Ɛ1, 4ƐƐ, 507, 511, 517, 51Ɛ, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58Ɛ, 591, 59Ɛ, 5Ɛ1, 5Ɛ5, 5Ɛ7, 5ƐƐ, 611, 615, 617, 61Ɛ, 637, 63Ɛ, 647, 655, 661, 665, 66Ɛ, 675, 687, 68Ɛ, 695, 69Ɛ, 6ᘔ7, 6Ɛ1, 701, 705, 70Ɛ, 711, 71Ɛ, 721, 727, 735, 737, 745, 747, 751, 767, 76Ɛ, 771, 775, 77Ɛ, 785, 791, 797, 7ᘔ1, 7ƐƐ, 801, 80Ɛ, 817, 825, 82Ɛ, 835, 841, 851, 855, 85Ɛ, 865, 867, 871, 881, 88Ɛ, 8ᘔ5, 8ᘔ7, 8ᘔƐ, 8Ɛ5, 8Ɛ7, 901, 905, 907, 90Ɛ, 91Ɛ, 921, 927, 955, 95Ɛ, 965, 971, 987, 995, 9ᘔ7, 9ᘔƐ, 9Ɛ1, 9Ɛ5, 9ƐƐ, ᘔ07, ᘔ0Ɛ, ᘔ11, ᘔ17, ᘔ27, ᘔ35, ᘔ37, ᘔ3Ɛ, ᘔ41, ᘔ45, ᘔ4Ɛ, ᘔ5Ɛ, ᘔ6Ɛ, ᘔ77, ᘔ87, ᘔ91, ᘔ95, ᘔ9Ɛ, ᘔᘔ7, ᘔᘔƐ, ᘔƐ7, ᘔƐƐ, Ɛ11, Ɛ15, Ɛ1Ɛ, Ɛ21, Ɛ25, Ɛ2Ɛ, Ɛ31, Ɛ37, Ɛ45, Ɛ61, Ɛ67, Ɛ6Ɛ, Ɛ71, Ɛ91, Ɛ95, Ɛ97, Ɛᘔ5, ƐƐ5, ƐƐ7

Except 2 and 3, all primes end with 1, 5, 7 or Ɛ. The first k such that all of 10k, 10k + 1, 10k + 2, ..., 10k + Ɛ are composite is 38, i.e. all of 380, 381, 382, ..., 38Ɛ are composite. (numbers k such that all of 10k, 10k + 1, 10k + 2, ..., 10k + Ɛ are composite are 38, 5ᘔ, 60, 62, 89, 93, 94, Ɛ0, Ɛ5, Ɛ8, ...)

The maximal gaps of primes less than 1000 is (927, 955), the gap (the difference of these two numbers) is 2ᘔ (i.e. there are 29 consecutive composites between 927 and 955).

The density of primes end with 1 is a relatively low (< 1/4), but the density of primes end with 5, 7 and Ɛ are nearly equal (all are a little more than 1/4). (i.e. for a given natural number N, the number of primes end with 1 less than N is usually smaller than the number of primes end with 5 (or 7, or Ɛ) less than N) e.g. For all 1426 primes < 10000, there are 3Ɛ8 primes (2Ɛ.3%) end with 1, 410 primes (30.3%) end with 5, 412 primes (30.5%) end with 7, 406 primes (2Ɛ.Ɛ%) end with Ɛ. It is conjectured that for every natural number N ≥ 10, the number of primes end with 1 less than N is smaller than the number of primes end with 5 (or 7, or Ɛ) less than N. (Note: the percentage in this section are also in duodecimal, i.e. 20% means 0.2 or 20/100 = 1/6, 36% means 0.36 or 36/100 = 7/20, 58.7% means 0.587 or 587/1000)

13665 is the smallest prime p such that the number of primes end with 1 or 5 ≤ p is more than the number of primes end with 3, 7 or Ɛ ≤ p (see (sequence A007350 in the OEIS), of course, 3 is the only prime ends with 3). Besides, 9ᘔ03693ᘔ831 is the smallest prime p such that the number of primes end with 1 or 7 ≤ p is more than the number of primes end with 2, 5 or Ɛ ≤ p (see (sequence A007352 in the OEIS), of course, 2 is the only prime ends with 2). Question: What is the smallest prime p such that the number of primes end with 1 ≤ p is more than the number of primes end with d ≤ p for at least one of d = 5, 7 or Ɛ?

All squares of primes (except 2 and 3) end with 1.

There are 2ᘔ primes between 0 and 100, 23 primes between 100 and 200, 1ᘔ primes between 200 and 300, 1ᘔ primes between 300 and 400, 1Ɛ primes between 400 and 500, 1ᘔ primes between 500 and 600, 16 primes between 600 and 700, 1ᘔ primes between 700 and 800, 18 primes between 800 and 900, 16 primes between 900 and ᘔ00, 1ᘔ primes between ᘔ00 and Ɛ00, 17 primes between Ɛ00 and 1000.

There are about N/ln(N) primes less than N, where ln is the natural logarithm, i.e. the logarithm with base e = 2.875236069821..., thus there are about $\frac{10^n}{n \cdot ln(10)}$ primes less than 10ⁿ (i.e. with at most n digits), and ln(10) = 2.599Ɛ035Ɛ8169...

N		total numbers of primes less than N		numbers of primes end with 1 less than N		numbers of primes end with 5 less than N		numbers of primes end with 7 less than N		numbers of primes end with Ɛ less than N
Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.
10	12	5	5	0	0	1	1	1	1	1	1
40	48	13	15	2	2	4	4	4	4	3	3
100	144	2ᘔ	34	6	6	9	9	9	9	8	8
400	576	89	105	1ᘔ	22	23	27	23	27	23	27
1000	1728	1ᘔ5	269	51	61	59	69	59	69	58	68
4000	6912	621	889	157	211	16ᘔ	226	170	228	166	222
10000	20736	1426	2334	3Ɛ8	572	410	588	412	590	406	582
40000	82944	4833	8103	11ᘔ4	1996	121Ɛ	2039	1219	2037	1211	2029
100000	248832	10852	21950	31ᘔ4	5452	3225	5501	3225	5501	321ᘔ	5494
400000	995328	3928Ɛ	78155	Ɛ333	19479	Ɛ377	19531	Ɛ3Ɛ9	19581	Ɛ3ᘔ2	19562
1000000	2985984	ᘔ4Ɛ20	215880	27204	53860	27295	53969	2730ᘔ	54010	27333	54039

In the following table, numbers shaded in cyan are primes.

1	2	3	4	5	6	7	8	9	ᘔ	Ɛ	10
11	12	13	14	15	16	17	18	19	1ᘔ	1Ɛ	20
21	22	23	24	25	26	27	28	29	2ᘔ	2Ɛ	30
31	32	33	34	35	36	37	38	39	3ᘔ	3Ɛ	40
41	42	43	44	45	46	47	48	49	4ᘔ	4Ɛ	50
51	52	53	54	55	56	57	58	59	5ᘔ	5Ɛ	60
61	62	63	64	65	66	67	68	69	6ᘔ	6Ɛ	70
71	72	73	74	75	76	77	78	79	7ᘔ	7Ɛ	80
81	82	83	84	85	86	87	88	89	8ᘔ	8Ɛ	90
91	92	93	94	95	96	97	98	99	9ᘔ	9Ɛ	ᘔ0
ᘔ1	ᘔ2	ᘔ3	ᘔ4	ᘔ5	ᘔ6	ᘔ7	ᘔ8	ᘔ9	ᘔᘔ	ᘔƐ	Ɛ0
Ɛ1	Ɛ2	Ɛ3	Ɛ4	Ɛ5	Ɛ6	Ɛ7	Ɛ8	Ɛ9	Ɛᘔ	ƐƐ	100

The twin prime pairs below 100 are (3, 5), (5, 7), (Ɛ, 11), (15, 17), (25, 27), (35, 37), (4Ɛ, 51), (5Ɛ, 61), (85, 87), (8Ɛ, 91), (Ɛ5, Ɛ7), and there are only 47 twin prime pairs below 1000. Except (3, 5), all twin prime pairs end with (5, 7) or (Ɛ, 1), and it is conjectured that the density of them are the same.

Fractions and irrational numbers

Fractions

Duodecimal fractions may be simple:

• 12 = 0.6
• 13 = 0.4
• 14 = 0.3
• 16 = 0.2
• 18 = 0.16
• 19 = 0.14
• 110 = 0.1 (note that this is a twelfth, 1ᘔ is a tenth)
• 114 = 0.09 (note that this is a sixteenth, 112 is a fourteenth)

or complicated:

• 15 = 0.249724972497... recurring (rounded to 0.24ᘔ)
• 17 = 0.186ᘔ35186ᘔ35... recurring (rounded to 0.187)
• 1ᘔ = 0.1249724972497... recurring (rounded to 0.125)
• 1Ɛ = 0.111111111111... recurring (rounded to 0.111)
• 111 = 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)
• 112 = 0.0ᘔ35186ᘔ35186... recurring (rounded to 0.0ᘔ3)
• 113 = 0.0972497249724... recurring (rounded to 0.097)

Examples in duodecimal	Decimal equivalent
1 × (58) = 0.76	1 × (58) = 0.625
100 × (58) = 76	144 × (58) = 90
5769 = 76	8109 = 90
4009 = 54	5769 = 64
1ᘔ.6 + 7.6 = 26	22.5 + 7.5 = 30

As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: 18 = 1(2×2×2), 120 = 1(2×2×5) and 1500 = 1(2×2×5×5×5) can be expressed exactly as 0.125, 0.05 and 0.002 respectively. 13 and 17, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, 18 is exact; 120 and 1500 recur because they include 5 as a factor; 13 is exact; and 17 recurs, just as it does in decimal.

The number of denominators which give terminating fractions within a given number of digits, say n, in a base b is the number of factors (divisors) of bⁿ, the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of bⁿ is given using its prime factorization.

For decimal, 10ⁿ = 2ⁿ * 5ⁿ. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together. Factors of 10ⁿ = (n+1)(n+1) = (n+1)².

For example, the number 8 is a factor of 10³ (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625_ten

For duodecimal, 12ⁿ = 2²ⁿ * 3ⁿ. This has (2n+1)(n+1) divisors. The sample denominator of 8 is a factor of a gross (12² = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76_twelve

Because both ten and twelve have two unique prime factors, the number of divisors of bⁿ for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n²).

Recurring digits

The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life division problems than factors of 5. Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.

However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the composite number 9. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, whereas only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are powers of two will have a shorter, more convenient terminating representation in duodecimal than in decimal representation (e.g. 1/(2²) = 0.25 _ten = 0.3 _twelve; 1/(2³) = 0.125 _ten = 0.16 _twelve; 1/(2⁴) = 0.0625 _ten = 0.09 _twelve; 1/(2⁵) = 0.03125 _ten = 0.046 _twelve; etc.).

Values in bold indicate that value is exact.

Decimal base Prime factors of the base: 2, 5 Prime factors of one below the base: 3 Prime factors of one above the base: 11 All other primes: 7, 13, 17, 19, 23, 29, 31			Duodecimal base Prime factors of the base: 2, 3 Prime factors of one below the base: Ɛ Prime factors of one above the base: 11 All other primes: 5, 7, 15, 17, 1Ɛ, 25, 27
Fraction	Prime factors of the denominator	Positional representation	Positional representation	Prime factors of the denominator	Fraction
1/2	2	0.5	0.6	2	1/2
1/3	3	0.3	0.4	3	1/3
1/4	2	0.25	0.3	2	1/4
1/5	5	0.2	0.2497	5	1/5
1/6	2, 3	0.16	0.2	2, 3	1/6
1/7	7	0.142857	0.186ᘔ35	7	1/7
1/8	2	0.125	0.16	2	1/8
1/9	3	0.1	0.14	3	1/9
1/10	2, 5	0.1	0.12497	2, 5	1/ᘔ
1/11	11	0.09	0.1	Ɛ	1/Ɛ
1/12	2, 3	0.083	0.1	2, 3	1/10
1/13	13	0.076923	0.0Ɛ	11	1/11
1/14	2, 7	0.0714285	0.0ᘔ35186	2, 7	1/12
1/15	3, 5	0.06	0.09724	3, 5	1/13
1/16	2	0.0625	0.09	2	1/14
1/17	17	0.0588235294117647	0.08579214Ɛ36429ᘔ7	15	1/15
1/18	2, 3	0.05	0.08	2, 3	1/16
1/19	19	0.052631578947368421	0.076Ɛ45	17	1/17
1/20	2, 5	0.05	0.07249	2, 5	1/18
1/21	3, 7	0.047619	0.06ᘔ3518	3, 7	1/19
1/22	2, 11	0.045	0.06	2, Ɛ	1/1ᘔ
1/23	23	0.0434782608695652173913	0.06316948421	1Ɛ	1/1Ɛ
1/24	2, 3	0.0416	0.06	2, 3	1/20
1/25	5	0.04	0.05915343ᘔ0Ɛ62ᘔ68781Ɛ	5	1/21
1/26	2, 13	0.0384615	0.056	2, 11	1/22
1/27	3	0.037	0.054	3	1/23
1/28	2, 7	0.03571428	0.05186ᘔ3	2, 7	1/24
1/29	29	0.0344827586206896551724137931	0.04Ɛ7	25	1/25
1/30	2, 3, 5	0.03	0.04972	2, 3, 5	1/26
1/31	31	0.032258064516129	0.0478ᘔᘔ093598166Ɛ74311Ɛ28623ᘔ55	27	1/27
1/32	2	0.03125	0.046	2	1/28
1/33	3, 11	0.03	0.04	3, Ɛ	1/29
1/34	2, 17	0.02941176470588235	0.0429ᘔ708579214Ɛ36	2, 15	1/2ᘔ
1/35	5, 7	0.0285714	0.0414559Ɛ3931	5, 7	1/2Ɛ
1/36	2, 3	0.027	0.04	2, 3	1/30

The duodecimal period length of 1/n are

0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 in the OEIS)

The duodecimal period length of 1/(nth prime) are

0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 in the OEIS)

Smallest prime with duodecimal period n are

11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 in the OEIS)

Irrational numbers

Algebraic irrational number	In decimal	In duodecimal
√2 (the length of the diagonal of a unit square)	1.414213562373... (≈ 1.414)	1.4Ɛ79170ᘔ07Ɛ8... (≈ 1.4Ɛ7)
√3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side)	1.732050807568... (≈ 1.732)	1.894Ɛ97ƐƐ9687... (≈ 1.895)
√5 (the length of the diagonal of a 1×2 rectangle)	2.236067977499... (≈ 2.236)	2.29ƐƐ13254058... (≈ 2.2ᘔ)
φ (phi, the golden ratio = $\scriptstyle \frac{1+\sqrt{5}}{2}$)	1.618033988749... (≈ 1.618)	1.74ƐƐ6772802ᘔ... (≈ 1.75)
Transcendental irrational number	In decimal	In duodecimal
π (pi, the ratio of circumference to diameter)	3.141592653589793238462643... (≈ 3.142)	3.184809493Ɛ918664573ᘔ6211... (≈ 3.185)
e (the base of the natural logarithm)	2.718281828459... (≈ 2.718)	2.875236069821... (≈ 2.875)

The first few digits of the decimal and duodecimal representation of another important number, the Euler–Mascheroni constant (the status of which as a rational or irrational number is not yet known), are:

Number	In decimal	In duodecimal
γ (the limiting difference between the harmonic series and the natural logarithm)	0.577215664901... (≈ 0.577)	0.6Ɛ15188ᘔ6760... (≈ 0.6Ɛ1)

Gallery

Duodecimal clock

• Dozenal Clock by Joshua Harkey
• Dozenal Clock with four hands and a digital display, in several variants, by Paul Rapoport