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 n^{2} ends with 0, if n is not divisible by 2 or 3, then n^{2} ends with 1, if n is divisible by 2 but not by 3, then n^{2} ends with 4, if n is not divisible by 2 but by 3, then n^{2} ends with 9. If the unit digit of n^{2} is 0, then the dozens digit of n^{2} is either 0 or 3, if the unit digit of n^{2} is 1, then the dozens digit of n^{2} is even, if the unit digit of n^{2} is 4, then the dozen digit of n^{2} is 0, 1, 4, 5, 8 or 9, if the unit digit of n^{2} is 9, then the dozen digit of n^{2} 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 E.
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 X) system squares may end in 444, the smallest example is 32^{2} = X04 = 1444_{(X)})
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 11^{2} = 121, and the only four-digit palindromic number divisible by 121 are 1331, 2662, 3993, 5225, 6556, 7887, 8EE8, 9119, X44X and E77E, 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, 16E61, 40804, 41414, 44944 | 101, 111, 121, 12E, 131, 202, 204, 212 | 8 |
6 | 160061 | 42E | 1 |
7 | 1002001, 102X201, 1093901, 1234321, 148X841, 4008004, 445X544, 49XXX94 | 1001, 1015, 1047, 1111, 1221, 2002, 2112, 2244 | 8 |
8 | none | none | 0 |
9 | 100020001, 102030201, 104060401, 1060E0601, 121242121, 123454321, 125686521, 1420E0241, 1444X4441, 1468E8641, 14X797X41, 1621E1261, 163151361, 1XX222XX1, 400080004, 404090404, 410212014, 4414X4144, 4456E6544, 496787694, 963848369 | 10001, 10101, 10201, 10301, 11011, 11111, 11211, 11E21, 12021, 12121, 1229E, 1292E, 12977, 14685, 20002, 20102, 20304, 21012, 21112, 22344, 31053 | 19 |
X | 1642662461 | 434X5 | 1 |
E | 10000200001, 10221412201, 10444X44401, 12102420121, 12345654321, 141E1E1E141, 14404X40441, 16497679461, 40000800004, 40441X14404, 41496869414, 44104X40144, 49635653694 | 100001, 101101, 102201, 110011, 111111, 11E13E, 120021, 12X391, 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_{n}^{2} (where R_{n} is the repunit with length n) is a palindromic number for n ≤ E, but not for n ≥ 10 (thus, for all odd number n ≤ 19, ther is n-digit palindromic square 123...321), besides, 11^{n} (also 1{0}1^{n}, i.e. 101^{n}, 1001^{n}, 10001^{n}, 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 X (in fact, no perfect powers end with 2, 6 or X), if n is not congruent to 2 mod 4, then n^{3} ends with the same digit as n; if n is congruent to 2 mod 4, then n^{3} 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 n^{i} and n^{j} end with the same digit.
Squares (and every powers) of 0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, E3854, 1E3854, X08369, ... end with the same digits as the number itself. (since they are automorphic numbers, from the only four solutions of x^{2}−x=0 in the ring of 10-adic numbers, these solutions are 0, 1, ...2E21E61E3854 and ...909X05X08369, since 10 is neither a prime not a prime power, the ring of the 10-adic numbers is not a field, thus there are solutions other than 0 and 1 for this equation in 10-adic numbers)
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 X.
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.
Template:Diagonal split header | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1X | 1E | 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 |
X | 1 | X | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 |
E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 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 |
1X | 1 | X | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 |
1E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 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, E (all digits != 2 mod 4) |
Template:Diagonal split header | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1X | 1E | 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 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 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 | X5 | 41 | 85 | 61 | 65 | 81 | 45 | X1 | 25 | 01 | 05 | 21 | X5 | 41 | 85 | 61 | 65 | 81 | 45 | X1 | 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 |
X | 01 | 0X | 84 | E4 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 1 |
E | 01 | 0E | X1 | 2E | 81 | 4E | 61 | 6E | 41 | 8E | 21 | XE | 01 | 0E | X1 | 2E | 81 | 4E | 61 | 6E | 41 | 8E | 21 | XE | 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 | X1 | E1 | 01 | 11 | 21 | 31 | 41 | 51 | 61 | 71 | 81 | 91 | X1 | E1 | 01 | 10 |
12 | 01 | 12 | 44 | 08 | 94 | X8 | 54 | 28 | 14 | 68 | 94 | X8 | 54 | 28 | 14 | 68 | 94 | X8 | 54 | 28 | 14 | 68 | 94 | X8 | 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 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 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 |
1X | 01 | 1X | 44 | E4 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 3 |
1E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 1E | 81 | 5E | 41 | 9E | 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 10^{n−1} (= λ(10^{n})).
Template:Diagonal split header | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1X | 1E | 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 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X |
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 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X |
7 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X |
8 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X |
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 |
X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | 2 |
E | 1 | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | 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 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X |
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 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X |
16 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X |
17 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X |
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 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | 2 |
1X | 1 | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | 1 |
1E | 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 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X |
The period of the digital roots of powers of a number must be a divisor of X (= λ(E)).
n | possible digital root of an nth power |
---|---|
0 | 1 |
= 1, 3, 7, 9 (mod X) | any number |
= 2, 4, 6, 8 (mod X) | 1, 3, 4, 5, 9, E |
= 5 (mod X) | 1, X, E |
> 0 and divisible by X | 1, E |
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 10^{4}.
In the following table, F_{n} is the n-th Fibonacci number, and L_{n} is the n-th Lucas number.
n | F_{n} | digit root of F_{n} | L_{n} | digit root of L_{n} | n | F_{n} | digit root of F_{n} | L_{n} | digit root of L_{n} |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 21 | 37501 | 5 | 81101 | E |
2 | 1 | 1 | 3 | 3 | 22 | 5X301 | 8 | 111103 | 7 |
3 | 2 | 2 | 4 | 4 | 23 | 95802 | 2 | 192204 | 7 |
4 | 3 | 3 | 7 | 7 | 24 | 133E03 | X | 2X3307 | 3 |
5 | 5 | 5 | E | E | 25 | 209705 | 1 | 47550E | X |
6 | 8 | 8 | 16 | 7 | 26 | 341608 | E | 758816 | 2 |
7 | 11 | 2 | 25 | 7 | 27 | 54E111 | 1 | 1012125 | 1 |
8 | 19 | X | 3E | 3 | 28 | 890719 | 1 | 176X93E | 3 |
9 | 2X | 1 | 64 | X | 29 | 121E82X | 2 | 2780X64 | 4 |
X | 47 | E | X3 | 2 | 2X | 1XE0347 | 3 | 432E7X3 | 7 |
E | 75 | 1 | 147 | 1 | 2E | 310EE75 | 5 | 6XE0647 | E |
10 | 100 | 1 | 22X | 3 | 30 | 5000300 | 8 | E22022X | 7 |
11 | 175 | 2 | 375 | 4 | 31 | 8110275 | 2 | 16110875 | 7 |
12 | 275 | 3 | 5X3 | 7 | 32 | 11110575 | X | 25330XX3 | 3 |
13 | 42X | 5 | 958 | E | 33 | 1922082X | 1 | 3E441758 | X |
14 | 6X3 | 8 | 133E | 7 | 34 | 2X3311X3 | E | 6477263E | 2 |
15 | E11 | 2 | 2097 | 7 | 35 | 47551X11 | 1 | X3EE4197 | 1 |
16 | 15E4 | X | 3416 | 3 | 36 | 75882EE4 | 1 | 148766816 | 3 |
17 | 2505 | 1 | 54E1 | X | 37 | 101214X05 | 2 | 23075X9E1 | 4 |
18 | 3XE9 | E | 8907 | 2 | 38 | 176X979E9 | 3 | 379305607 | 7 |
19 | 6402 | 1 | 121E8 | 1 | 39 | 2780E0802 | 5 | 5X9X643E8 | E |
1X | X2EE | 1 | 1XE03 | 3 | 3X | 432E885EE | 8 | 967169X03 | 7 |
1E | 14701 | 2 | 310EE | 4 | 3E | 6XE079201 | 2 | 13550121EE | 7 |
20 | 22X00 | 3 | 50002 | 7 | 40 | E22045800 | X | 2100180002 | 3 |
(Note that F_{2X} begins with L_{1X}, and F_{2E} begins with L_{1E})
The period of the digit root of Fibonacci numbers is X.
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 2×10^{n−1} (n ≥ 2). (see Pisano period)
There are only 13 possible values (of the totally 100 values, thus only 13%) of the final two digits of a Fibonacci number (see Template:Oeis).
Except 0 = F_{0} and 1 = F_{1} = F_{2}, the only square Fibonacci number is 100 = F_{10} (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: F_{11}/F_{10} = 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 = F_{6}.
n | 2^{n} | n | 2^{n} | n | 2^{n} | n | 2^{n} | n | 2^{n} | n | 2^{n} |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 21 | E2X20X8 | 41 | 5317E5804588X8 | 61 | 256906X1X93096E8934X8 | 81 | 11X12X743504482569888538X0X8 | X1 | 65933E8691303X448E712227X7E11448X8 |
2 | 4 | 22 | 1X584194 | 42 | X633XE408E5594 | 62 | 4E161183966171E566994 | 82 | 238259286X08944E17554X758194 | X2 | 10E667E51626078895E22445393X2289594 |
3 | 8 | 23 | 38E48368 | 43 | 190679X815XXE68 | 63 | 9X302347710323XE11768 | 83 | 4744E6551815689X32XX992E4368 | X3 | 21E113XX305013556EX4488X76784556E68 |
4 | 14 | 24 | 75X94714 | 44 | 361137942E99E14 | 64 | 1786046932206479X23314 | 84 | 9289E0XX342E15786599765X8714 | X4 | 43X2279860X026XE1E88955931348XE1E14 |
5 | 28 | 25 | 12E969228 | 45 | 702273685E77X28 | 65 | 3350091664410937846628 | 85 | 16557X198685X2E350E7730E95228 | X5 | 878453750180519X3E556XE6626959X3X28 |
6 | 54 | 26 | 25E716454 | 46 | 120452714EE33854 | 66 | 66X0163108821673491054 | 86 | 30XE3837514E85X6X1E3261E6X454 | X6 | 15348X72X0340X3787XXE19E10516E787854 |
7 | X8 | 27 | 4EE2308X8 | 47 | 2408X5229EX674X8 | 67 | 111803062154431269620X8 | 87 | 619X7472X29E4E9183X6503E188X8 | X7 | 2X695925806818735399X37X20X31E3534X8 |
8 | 194 | 28 | 9EX461594 | 48 | 48158X457E912994 | 68 | 2234061042X886251704194 | 88 | 103792925857X9E634790X07X35594 | X8 | 5916E64E41143526X777873841863X6X6994 |
9 | 368 | 29 | 17E8902E68 | 49 | 942E588E3E625768 | 69 | 446810208595504X3208368 | 89 | 20736564E4E397E06936181386XE68 | X9 | E631E09X82286X5193335274835079191768 |
X | 714 | 2X | 33E5605E14 | 4X | 1685XE55X7E04E314 | 6X | 891420414E6XX0986414714 | 8X | 41270E09X9X773X116703427519E14 | XX | 1E063X179445518X36666X52946X136363314 |
E | 1228 | 2E | 67XE00EX28 | 4E | 314E9XXE93X09X628 | 6E | 1562840829E1981750829228 | 8E | 82521X179793278231206852X37X28 | XE | 3X107833688XX358711118X56918270706628 |
10 | 2454 | 30 | 1139X01E854 | 50 | 629E799E678179054 | 70 | 2E05481457X37432X1456454 | 90 | 144X4383373665344624114X5873854 | E0 | 78213467155986E52222358E1634521211054 |
11 | 48X8 | 31 | 2277803E4X8 | 51 | 1057E377E1343360X8 | 71 | 5X0X9428E3872865828E08X8 | 91 | 2898874672710X6890482298E5274X8 | E1 | 1344269122XE751XX44446E5X3068X424220X8 |
12 | 9594 | 32 | 4533407X994 | 52 | 20E3X733X268670194 | 72 | E8196855X752550E455X1594 | 92 | 557552912522191560944575XX52994 | E2 | 26885162459E2X39888891XE86115884844194 |
13 | 16E68 | 33 | 8X668139768 | 53 | 41X792678515120368 | 73 | 1E43714XE92X4XX1X8XE82E68 | 93 | XE2XX5624X44362E01688E2E98X5768 | E3 | 5154X3048E7X58775555639E5022E549488368 |
14 | 31E14 | 34 | 159114277314 | 54 | 839365134X2X240714 | 74 | 3X872299E6589983959E45E14 | 94 | 19X598E049888705X03155X5E758E314 | E4 | X2X986095E38E532XXXE077XX045XX96954714 |
15 | 63X28 | 35 | 2E6228532628 | 55 | 147670X269858481228 | 75 | 79524577E0E577476E7X8EX28 | 95 | 378E75X09755520E8062XE8EE2E5X628 | E5 | 185975016EX75XX65999X1339808E99716X9228 |
16 | 107854 | 36 | 5E0454X65054 | 56 | 29312185174E4942454 | 76 | 136X48E33X1XE32931E395E854 | 96 | 735E2E8172XXX41E41059E5EX5XE9054 | E6 | 34E72X031E92E990E77782677415E7723196454 |
17 | 2134X8 | 37 | EX08X990X0X8 | 57 | 5662434X329X96848X8 | 77 | 271895X67839X65663X76EE4X8 | 97 | 126EX5E432599883X820E7XEE8E9E60X8 | E7 | 69E258063E65E761E3334513282EE32463708X8 |
18 | 426994 | 38 | 1E81597618194 | 58 | E104869865797149594 | 78 | 52356E91347790E107931EX994 | 98 | 251E8EX864E775479441E39EE5E7E0194 | E8 | 117X4E4107E0EE303X6668X26545EX6490721594 |
19 | 851768 | 39 | 3E42E73034368 | 59 | 1X20951750E372296E68 | 79 | X46E1E62693361X213663E9768 | 99 | 4X3E5E9509E32X936883X77EXEE3X0368 | E9 | 23389X8213X1EX60791115850X8EE90961242E68 |
1X | 14X3314 | 3X | 7X85E26068714 | 5X | 38416X32X1X724571E14 | 7X | 1891X3E051667038427107E7314 | 9X | 987XEE6X17X659671547933E9EX780714 | EX | 467579442783E90136222E4X195EE61702485E14 |
1E | 2986628 | 3E | 1394EX50115228 | 5E | 74831865839248E23X28 | 7E | 356387X0X3112074852213E2628 | 9E | 17539EE183390E7122X93667E7E9341228 | EE | 912E36885347E60270445X9836EEE0320494EX28 |
20 | 5751054 | 40 | 2769E8X022X454 | 60 | 12946350E476495X47854 | 80 | 6E075381862241294X4427X5054 | X0 | 32X77EX346761E2245967113E3E6682454 | 100 | 1625X7154X693E0052088E97471EEX0640969E854 |
The smallest power of 2 starts with the digit E is 2^{21} = E2X20X8. For all digits 1 ≤ d ≤ E, there exists 0 ≤ n ≤ 21 such that 2^{n} starts with the digit d.
2^{1XE} = 59E18922E81631X39875663E89X853X91E595336X6114815X5X6929933X288E774E479575X628 may be the largest power of 2 not contain the digit 0, it has 65 digits. (Note that in the decimal (base X) system, the largest power of 2 not contain the digit 0 is 2^{72} = E8196855X752550E455X1594 = 77371252455336267181195264_{X}, it has only 22 digits in base X)
The number 2^{29} = 2^{368} (see power of 2#Powers of two whose exponents are powers of two) is very close to googol (10^{100}), since it has EE digits. (thus, the Fermat number F_{9} (=2^{29}+1) is very close to googol)
1001 is the first four-digit palindromic number, and it is also the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=1^{3} + 10^{3}) = 509 + 6E4 (=9^{3} + X^{3}) (see taxicab number for other numbers), and it is also the smallest absolute Euler pseudoprime, note that there is no absolute Euler-Jacobi pseudoprime and no absolute strong pseudoprime. Since 1001 = 7×11×17, we can use the divisibility rule of 1001 (i.e. form the alternating sum of blocks of three from right to left) for the divisibility rule of 7, 11 and 17. Besides, if 6k+1, 10k+1 and 16k+1 are all primes, then the product of them must be a Carmichael number (absolute Fermat pseudoprime), the smallest case is indeed 1001 (for k = 1), but 1001 is not the smallest Carmichael number (the smallest Carmichael number is 3X9).
$ \sqrt{2} $ is very close to 1.5, since a near-value for $ \sqrt{2} $ is 15/10 (=N_{4}/P_{4}, where N_{n} is nth NSW number, and P_{n} is nth Pell number, N_{n}/P_{n} is very close to $ \sqrt{2} $ when n is large). Besides, $ \sqrt{5} $ is very close to 2.2X, since a near-value for $ \sqrt{5} $ is 22X/100 (= L_{10}/F_{10}, where L_{n} is nth Lucas number, and F_{n} is nth Fibonacci number, L_{n}/F_{n} is very close to $ \sqrt{5} $ when n is large).
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, X8, 100, 116, 140, 160, 183, 194, 200, 230, 280, 300, 346, 368, 400, 460, 509, 540, 600, 690, 714, 800, 900, X16, X80, 1000.
If and only if n is a divisor of 20, then m^{2} = 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 E), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or E (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 relatively low, but the density of primes end with 5, 7 and E are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or E)
Except (3, 5), all twin primes end with (5, 7) or (E, 1), and the density of these two types of twin primes are nearly equal.
The sum of any pair of twin primes (other than (3, 5)) ends with 0.
If n ≥ 3 and n is not divisible by E, 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, E, 11, 111, 131, 141, 171, 181, 1E1, 535, 545, 565, 575, 585, 5E5, 727, 737, 747, 767, 797, E1E, E2E, E6E.
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 E.
Except for 5 and 7, all safe primes end with E.
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 E (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 E (or p=2).
A prime p can be written as x^{2} + y^{2} if and only if p ends with 1 or 5 (or p=2).
A prime p can be written as x^{2} + 3y^{2} if and only if p ends with 1 or 7 (or p=3).
All full reptend primes end with 5 or 7. (in fact, for all primes p ≥ 5, (p-1)/(the period length of 1/p) is odd if and only if p is end with 5 or 7, since 10 is a quadratic nonresidue mod p (i.e. $ \left(\frac{10}{p}\right)=-1 $, where $ \left(\frac{m}{n}\right) $ is the Legendre symbol) if and only if p is end with 5 or 7, by quadratic reciprocity, and if 10 is a quadratic residue mod a prime, then 10 cannot be a primitive root mod this prime) However, the converse is not true, 17 is not a full reptend prime, since the recurring digits of 1/17 is 0.076E45076E45..., which has only period 6. If and only if p is a full reptend prime, then the recurring digits of 1/p is cyclic number, e.g. the recurring digits of 1/5 is the cyclic number 2497 (the cyclic permutations of the digits are this number multiplied by 1 to 4), and the recurring digits of 1/7 is the cyclic number 186X35 (the cyclic permutations of the digits are this number multiplied by 1 to 6). The full reptend primes below 1000 are 5, 7, 15, 27, 35, 37, 45, 57, 85, 87, 95, X7, E5, E7, 105, 107, 117, 125, 145, 167, 195, 1X5, 1E5, 1E7, 205, 225, 255, 267, 277, 285, 295, 315, 325, 365, 377, 397, 3X5, 3E5, 3E7, 415, 427, 435, 437, 447, 455, 465, 497, 4X5, 517, 527, 535, 545, 557, 565, 575, 585, 5E5, 615, 655, 675, 687, 695, 6X7, 705, 735, 737, 745, 767, 775, 785, 797, 817, 825, 835, 855, 865, 8E5, 8E7, 907, 927, 955, 965, 995, 9X7, 9E5, X07, X17, X35, X37, X45, X77, X87, X95, XE7, E25, E37, E45, E95, E97, EX5, EE5, EE7. (Note that for the primes end with 5 or 7 below 30 (5, 7, 15, 17, 25 and 27, all numbers end with 5 or 7 below 30 are primes), 5, 7, 15 and 27 are full reptend primes, and since 5×25 = 101 = $ \Phi_4(10) $, the period of 25 is 4, which is the same as the period of 5, and we can use the test of the divisiblity of 5 to test that of 25 (form the alternating sum of blocks of two from right to left), and since 7×17 = E1 = $ \Phi_6(10) $, the period of 17 is 6, which is the same as the period of 7, and we can use the test of the divisiblity of 7 to test that of 17 (form the alternating sum of blocks of three from right to left), thus, 17 and 25 are not full reptend primes, and they are the only two non-full reptend primes end with 5 or 7 below 30)
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} = E for every 1 ≤ i ≤ n/2. e.g. 1/5 = 0.249724972497..., and 24 + 97 = EE, and 1/7 = 0.186X35186X35..., and 186 + X35 = EEE, all primes (other than 2 and 3) ≤ 37 except E, 1E and 31 have even period length, thus they can use Midy theorem to get an E-repdigit number, the length of this number is the period length of this prime. (see below for the recurring digits for 1/n for all n ≤ 30)
The unique primes below 10^{60} are E, 11, 111, E0E1, EE01, 11111, 24727225, E0E0E0E0E1, E00E00EE0EE1, 100EEEXEXEE000101, 1111111111111111111, EEEE0000EEEE0000EEEE0001, 100EEEXEE0000EEEXEE000101, 10EEEXXXE011110EXXXE00011, EEEEEEEE00000000EEEEEEEE00000001, EEE000000EEE000000EEEEEE000EEEEEE001, and the period length of their reciprocals are 1, 2, 3, X, 10, 5, 18, 1X, 19, 50, 17, 48, 70, 5X, 68, 53.
If p is a safe prime other than 5, 7 and E, then the period length of 1/p is (p-1)/2. (this is not true for all primes ends with E (other than E itself), the first counterexample is p = 2EE, 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 X) system there are some such primes, and may be infinitely many such primes, the first few such primes in that base are 61_{(X)} = 51, 131_{(X)} = XE, 181_{(X)} = 131, 461_{(X)} = 325, 491_{(X)} = 34E, see Template: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, ..., E appears in the repeating sequence the same number of times as does each other digit (namely, (p−1)/10 times)), see repeating decimal#Fractions with prime denominators) (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 E, and 10 is quadratic residue for all primes end with E. (In contrast, in decimal (base X) system there may be infinitely many such primes, the first few such primes in that base are 7_{(X)} = 7, 23_{(X)} = 1E, 47_{(X)} = 3E, 59_{(X)} = 4E, 167_{(X)} = 11E, see Template:Oeis) (if so, then this prime p produces a stream of p−1 pseudo-random digits, see repeating decimal#Fractions with prime denominators) (In fact, not only for base 10 there are only finitely many such primes, of course for square bases (bases of the form k^{2}) 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 3k^{2}, 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 | 41E | 20E | 591 | 3X | 767 | 766 | 927 | 926 | E1E | 56E | 1107 | 1106 | 12E5 | 12E4 | 14E1 | 14E | 16X7 | 16X6 |
3 | 0 | 117 | 116 | 271 | 27 | 421 | 63 | 59E | 2XE | 76E | 395 | 955 | 954 | E21 | 570 | 1115 | 1114 | 1301 | 760 | 14E5 | 14E4 | 16E5 | 188 |
5 | 4 | 11E | 6E | 277 | 276 | 427 | 426 | 5E1 | 159 | 771 | 132 | 95E | 48E | E25 | E24 | 1125 | 1124 | 1317 | 506 | 14EE | 85E | 16E7 | 16E6 |
7 | 6 | 125 | 124 | 27E | 13E | 431 | 109 | 5E5 | 5E4 | 775 | 774 | 965 | 964 | E2E | 575 | 112E | 675 | 1337 | 512 | 150E | 865 | 1705 | 398 |
E | 1 | 12E | 75 | 285 | 284 | 435 | 434 | 5E7 | 66 | 77E | 39E | 971 | 172 | E31 | 116 | 1135 | 1134 | 133E | 77E | 1517 | 1XX | 1711 | 493 |
11 | 2 | 131 | 76 | 291 | 83 | 437 | 436 | 5EE | 2EE | 785 | 784 | 987 | 32X | E37 | E36 | 114E | 685 | 1345 | 1344 | 1521 | 436 | 1715 | 1714 |
15 | 14 | 13E | 7E | 295 | 294 | 447 | 446 | 611 | 163 | 791 | 3X6 | 995 | 994 | E45 | E44 | 1151 | 115 | 1351 | 166 | 1525 | 1524 | 1727 | 1726 |
17 | 6 | 141 | 20 | 2X1 | 150 | 455 | 454 | 615 | 614 | 797 | 796 | 9X7 | 9X6 | E61 | 16 | 1165 | 1164 | 1365 | 1364 | 1547 | 1E2 | 1735 | 1734 |
1E | E | 145 | 144 | 2XE | 155 | 457 | 15X | 617 | 206 | 7X1 | 138 | 9XE | 4E5 | E67 | 3X2 | 1167 | 42 | 1367 | 1366 | 1561 | 89 | 1745 | 1744 |
25 | 4 | 147 | 56 | 2E1 | 26 | 45E | 22E | 61E | 30E | 7EE | 3EE | 9E1 | 9E | E6E | 595 | 1185 | 1184 | 136E | 795 | 156E | 97 | 1747 | 1746 |
27 | 26 | 157 | 12 | 2EE | 37 | 465 | 464 | 637 | 212 | 801 | 140 | 9E5 | 9E4 | E71 | 596 | 118E | 6X5 | 1377 | 106 | 1577 | 1576 | 1751 | 32X |
31 | 9 | 167 | 166 | 301 | 90 | 46E | 7 | 63E | 31E | 80E | 405 | 9EE | 4EE | E91 | 2E3 | 1197 | 472 | 138E | 7X5 | 157E | 89E | 1755 | 1754 |
35 | 34 | 16E | 95 | 307 | 102 | 471 | 13 | 647 | 216 | 817 | 816 | X07 | X06 | E95 | E94 | 11X1 | 6E | 1391 | 3E3 | 1585 | 1584 | 1757 | 1756 |
37 | 36 | 171 | 96 | 30E | 165 | 481 | 24 | 655 | 654 | 825 | 824 | X0E | 505 | E97 | E96 | 11X5 | 11X4 | 1395 | 1394 | 1587 | 2X | 176E | 995 |
3E | 1E | 175 | 8 | 315 | 314 | 485 | 44 | 661 | 176 | 82E | 415 | X11 | 56 | EX5 | EX4 | 11X7 | 11X6 | 13X1 | 7E0 | 1591 | 2E6 | 1781 | 4E0 |
45 | 44 | 17E | 9E | 321 | 170 | 48E | 245 | 665 | 138 | 835 | 834 | X17 | X16 | EE5 | EE4 | 11XE | 6E5 | 13X7 | 536 | 15XE | 8E5 | 1785 | 1784 |
4E | 25 | 181 | X0 | 325 | 324 | 497 | 496 | 66E | 335 | 841 | 84 | X27 | 156 | EE7 | EE6 | 11E7 | 11E6 | 13E1 | 13E | 15EE | 8EE | 178E | 9X5 |
51 | 13 | 18E | X5 | 327 | 10X | 4X5 | 4X4 | 675 | 674 | 851 | 14X | X35 | X34 | 1005 | 1004 | 1201 | 700 | 13E5 | 13E4 | 1601 | 160 | 1797 | 1796 |
57 | 56 | 195 | 194 | 32E | 175 | 4E1 | 9X | 687 | 686 | 855 | 854 | X37 | X36 | 1011 | 73 | 120E | 705 | 1405 | 1404 | 1615 | 1614 | 17X1 | 9E0 |
5E | 2E | 19E | XE | 33E | 17E | 4EE | 25E | 68E | 345 | 85E | 42E | X3E | 51E | 1017 | 1016 | 1211 | 706 | 1407 | 326 | 1621 | 910 | 17X5 | 17X4 |
61 | 30 | 1X5 | 1X4 | 347 | 46 | 507 | 182 | 695 | 694 | 865 | 864 | X41 | 188 | 1021 | 610 | 121E | 70E | 1425 | 1424 | 1625 | 1624 | 17EE | 9EE |
67 | 22 | 1X7 | 46 | 34E | 2E | 511 | 266 | 69E | 34E | 867 | 2X2 | X45 | X44 | 1027 | 1026 | 1231 | 123 | 142E | 815 | 1635 | 274 | 1807 | 682 |
6E | 35 | 1E1 | E6 | 357 | 11X | 517 | 516 | 6X7 | 6X6 | 871 | 152 | X4E | 525 | 1041 | 620 | 123E | 71E | 1431 | 286 | 1647 | 276 | 1815 | 1814 |
75 | 8 | 1E5 | 1E4 | 35E | 18E | 51E | 45 | 6E1 | 6E | 881 | 440 | X5E | 52E | 1047 | 1046 | 1245 | 1244 | 1437 | 1436 | 1655 | 1654 | 181E | X0E |
81 | 14 | 1E7 | 1E6 | 365 | 364 | 527 | 526 | 701 | 360 | 88E | 445 | X6E | 535 | 104E | 625 | 1255 | 114 | 143E | 81E | 1657 | 61X | 1825 | 1824 |
85 | 84 | 205 | 204 | 375 | 34 | 531 | 276 | 705 | 704 | 8X5 | 98 | X77 | X76 | 1051 | 313 | 1257 | 49X | 1445 | 1444 | 165E | 92E | 1831 | 509 |
87 | 86 | 217 | 86 | 377 | 376 | 535 | 534 | 70E | 365 | 8X7 | 2E6 | X87 | X86 | 1061 | 16 | 125E | 72E | 1457 | 1456 | 1667 | 622 | 183E | X1E |
8E | 45 | 21E | 10E | 391 | 1X6 | 541 | 54 | 711 | 71 | 8XE | 455 | X91 | 283 | 106E | 635 | 1261 | 730 | 1461 | 38 | 1671 | 936 | 184E | X25 |
91 | 46 | 221 | 66 | 397 | 396 | 545 | 544 | 71E | 36E | 8E5 | 8E4 | X95 | X94 | 107E | 63E | 126E | 735 | 1465 | 1464 | 1677 | 20X | 1861 | 269 |
95 | 94 | 225 | 224 | 3X5 | 3X4 | 557 | 556 | 721 | 370 | 8E7 | 8E6 | X9E | 54E | 1087 | 1086 | 127E | 73E | 1467 | 562 | 167E | 93E | 1865 | 1864 |
X7 | X6 | 237 | 92 | 3XE | 1E5 | 565 | 564 | 727 | 24X | 901 | 230 | XX7 | 376 | 109E | 64E | 1281 | 740 | 1471 | 419 | 1681 | 140 | 186E | X35 |
XE | 55 | 241 | 120 | 3E5 | 3E4 | 575 | 574 | 735 | 734 | 905 | 198 | XXE | 555 | 10E1 | 329 | 1295 | 94 | 1475 | 1474 | 1685 | 1684 | 1875 | 1874 |
E5 | E4 | 24E | 125 | 3E7 | 3E6 | 577 | 116 | 737 | 736 | 907 | 906 | XE7 | XE6 | 10E7 | 10E6 | 1297 | 1296 | 147E | 83E | 168E | 945 | 1877 | 146 |
E7 | E6 | 251 | 73 | 401 | 100 | 585 | 584 | 745 | 744 | 90E | 465 | XEE | 55E | 10EE | 65E | 12X1 | 75 | 148E | 845 | 1697 | 1696 | 189E | X4E |
105 | 104 | 255 | 254 | 40E | 205 | 587 | 1XX | 747 | 9X | 91E | 46E | E11 | 1X2 | 1101 | 220 | 12X5 | 2E8 | 1495 | 1494 | 169E | 94E | 18X1 | 210 |
107 | 106 | 25E | 12E | 415 | 414 | 58E | 2X5 | 751 | 1X3 | 921 | 236 | E15 | 228 | 1105 | 1104 | 12X7 | 12X6 | 149E | 84E | 16X1 | 950 | 18XE | X55 |
period length | primes | period length | primes |
---|---|---|---|
1 | E | 11 | 1E0411, 69X3901 |
2 | 11 | 12 | 157, 7687 |
3 | 111 | 13 | 51, 471, 57E1 |
4 | 5, 25 | 14 | 15, 81, 106X95 |
5 | 11111 | 15 | X9X9XE, 126180EE0EE |
6 | 7, 17 | 16 | E61, 1061 |
7 | 46E, 2X3E | 17 | 1111111111111111111 |
8 | 75, 175 | 18 | 24727225 |
9 | 31, 3X891 | 19 | E00E00EE0EE1 |
X | E0E1 | 1X | E0E0E0E0E1 |
E | 1E, 754E2E41 | 1E | 3E, 78935EX441, 523074X3XXE |
10 | EE01 | 20 | 141, 8E5281 |
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: 076E45 (for a = 1, 7, 8, E, 10, 16), 131X8X (for a = 2, 3, 5, 12, 14, 15) and 263958 (for a = 4, 6, 9, X, 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 08579214E36429X7.
There are only 9 repunit primes below R_{1000}: R_{2}, R_{3}, R_{5}, R_{17}, R_{81}, R_{91}, R_{225}, R_{255} and R_{4X5} (R_{n} is the repunit with length n). If p is a Sophie Germain prime other than 2, 3 and 5, then R_{p} is divisible by 2p+1, thus R_{p} is not prime. (The length for the repunit (probable) primes are 2, 3, 5, 17, 81, 91, 225, 255, 4X5, 5777, 879E, 198E1, 23175, 311407, ..., note that 879E is the smallest (and the only known) such number ends with E)
By Fermat's little theorem, if p is a prime other than 2, 3 and E, then p divides the repunit with length p−1. (The converse is not true, the first counterexample is 55, which is composite (equals 5×11) but divides the repunit with length 54, the counterexamples up to 1000 are 55, 77, E1, 101, 187, 275, 4X7, 777, 781, E55, they are exactly the Fermat pseudoprimes for base 10 (composite numbers c such that 10^{c-1} = 1 mod c) which are not divisible by E, they are called "deceptive primes", if n is deceptive prime, then R_{n} is also deceptive prime, thus there are infinitely may deceptive primes) Thus, we can prove that every positive integer coprime to 10 has a repunit multiple, and every positive integer has a multiple uses only 0's and 1's.
n | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +X | +E | +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 |
X0+ | 111111111101 | 100101010 | 1110010 | 1010110 | 11100001 | 100100 | 1100001 | 10000 | 1010010 | 11110 | 1111011111 | 111111111110 |
E0+ | 1001 | 11001010 | 101000 | 1011100 | 101011 | 111010 | 11010111 | 1011010 | 100011110 | 11001110 | 1111111111111111111111 | 100 |
n | 1 | 5 | 7 | E | 11 | 15 | 17 | 1E | 21 | 25 | 27 | 2E |
smallest k such that k×n is a repunit | 1 | 275 | 1X537 | 123456789E | 1 | 92X79E43715865 | 8327 | 69E63848E | 634X159788253X72E1 | 55 | 509867481E793XX5X1243628E317 | 45X3976X7E |
the length of the repunit k×n | 1 | 4 | 6 | E | 2 | 14 | 6 | E | 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, 19E, 275 and 46E, and only 19E and 46E 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 = R_{4} = 1111 = 5×11×25, besides, this k for n = 46E is 2X3E, which makes k×n = R_{7} = 1111111, a repunit semiprime, and this k for n = 19E is a X8-digit prime number, with k×n = R_{XE}, another repunit semiprime)
For every prime p except E, 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, E1, 101, 187, 1E0, 275, 444, 4X7, 777, 781, E55, 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 E. (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 E) 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 E, 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 E (of course, if n is divisible by E, then such number does not exist).
Smallest multiple of n with digit sum n are:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 1E0, 20E, 22X, 249, 268, 287, 2X6, 45X, 488, 4E6, 1EX, 8E4, 3EX0, 3EE, 23EX, 1899, XX8, 2E79, 4E96, 1EX9, 4XX8, 2EE9, 3XEX, 799X, 5EE90, ..., such number is conjectured to exist for all n.
45 is the smallest prime that produces prime reciprocal magic square, i.e. write the recurring digits of 1/45 (=0.Template:Overline, which has period 44) to 44/45, we get a 44×44 prime reciprocal magic square (its magic number is 1EX), it is conjectured that there are infinitely many such primes, but 45 is the only such prime below 1000, all such primes are full reptend primes, i.e. the reciprocal of them are cyclic numbers, and 10 is a primitive root modulo these primes.
All numbers of the form 34{1} are composite (proof: 34{1_{n}} = 34×10^{n}+(10^{n}−1)/E = (309×10^{n}−1)/E and it can be factored to ((19×10^{n/2}−1)/E) × (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_{E78}}, it has E7X 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 (see generalized Fermat prime), 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 EX, since all numbers obtained as the concatenation of a power of EX (>1) followed by a 1 are divisible by either E 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^{2781E5}+1, it has 459655 digits.
All numbers of the form 1{5}1 are composite (proof: 1{5_{n}}1 = (14×10^{n+1}−41)/E and it can be factored to (4×10^{(n+1)/2}−7) × ((4×10^{(n+1)/2}−7)/E) for odd n and divisible by 11 for even n).
The emirps below 1000 are 15, 51, 57, 5E, 75, E5, 107, 117, 11E, 12E, 13E, 145, 157, 16E, 17E, 195, 19E, 1X7, 1E5, 507, 51E, 541, 577, 587, 591, 59E, 5E1, 5EE, 701, 705, 711, 751, 76E, 775, 785, 7X1, 7EE, E11, E15, E21, E31, E61, E67, E71, E91, E95, EE5, EE7.
The non-repdigit permutable primes below 10^{10100} are 15, 57, 5E, 117, 11E, 5EEE (the smallest representative prime of the permutation set).
The non-repdigit circular primes below 10^{10100} are 15, 57, 5E, 117, 11E, 175, 1E7, 157E, 555E, 115E77 (the smallest representative prime of the cycle).
The first few Smarandache primes are the concatenation of the first 5, 15, 4E, 151, ... positive integers.
The only known Smarandache–Wellin primes are 2 and 2357E11.
There are exactly 15 minimal primes, and they are 2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001.
The smallest weakly prime is 6E8XE77.
The largest left-truncatable prime is 28-digit 471X34X164259EX16E324XE8X32E7817, and the largest right-truncatable prime is X-digit 375EE5E515.
The only two base 10 Wieferich primes up to 10^{10} are 1685 and 5E685, 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 (= 2^{3}) and 14 (= 2^{4}), 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, X, E, 25, X5, 577, 668, X83, 14765, 938X4, 369862, X2394X, ..., the largest of which is 43-digit 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E. (see Template: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, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {68, 84}, or one of the only two fixed points other than 1: 25 and X5. (In contrast, in the decimal (base X) system there are EX happy numbers below 1000_{(X)} (=6E4), 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, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}. (In contrast, in the decimal (base X) system all multiple of 3 get to 153_{(X)} (=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 = 9^{3} + X^{3}, we know that this sequence with initial term 9X is 9X, 1001, 2, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 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}, {X}, {E} | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 |
2 | {1}, {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {25}, {68, 84}, {X5} | 1, 2, X, 3, 1, 2, 1 |
3 | {1}, {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}, {577}, {668}, {6E5, E74, 100X}, {X83}, {11XX} | 1, 12, 1, 1, 3, 1, 1 |
4 | {1}, {X6X, 103X8, 8256, 35X9, 9EXE, 22643, E69, 1102X, 596X, X842, 8394, 6442, 1080, 2455}, {206X, 6668, 4754}, {3X2E, 12396, 472E, X02X, E700, 9X42, 98X9, 13902} | 1, 12, 3, 8 |
The harshad numbers up to 200 are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 1X, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, X0, X1, E0, 100, 10X, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1X0, 1E0, 1EX, 200, although the sequence of factorials begins with harshad numbers, not all factorials are harshad numbers, after 7! (=2E00, with digit sum 11 but 11 does not divide 7!), 8X4! is the next that is not (8X4! has digit sum 8275 = E×8E7, thus not divide 8X4!). 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, E, 56, 66, EE, 444, 778, EEE, 12XX, 1640, 2046, 2929, 3333, 4973, 5E60, 6060, 7249, 8889, 9293, 9E76, X580, X912, EEEE.
The Kaprekar's routine of any four-digit number which is not repdigit converges to either the cycle {3EE8, 8284, 6376} or the cycle {4198, 8374, 5287, 6196, 7EE4, 7375}, and the Kaprekar map of any three-digit number which is not repdigit converges to the fixed point 5E6, and the Kaprekar map of any two-digit number which is not repdigit converges to the cycle {0E, X1, 83, 47, 29, 65}.
n | cycles for Kaprekar's routine for n-digit numbers | length of these cycles |
---|---|---|
1 | {0} | 1 |
2 | {00}, {0E, X1, 83, 47, 29, 65} | 1, 6 |
3 | {000}, {5E6} | 1, 1 |
4 | {0000}, {3EE8, 8284, 6376}, {4198, 8374, 5287, 6196, 7EE4, 7375} | 1, 3, 6 |
5 | {00000}, {64E66, 6EEE5}, {83E74} | 1, 2, 1 |
6 | {000000}, {420X98, X73742, 842874, 642876, 62EE86, 951963, 860X54, X40X72, X82832, 864654}, {65EE56} | 1, X, 1 |
The self numbers up to 600 are 1, 3, 5, 7, 9, E, 20, 31, 42, 53, 64, 75, 86, 97, X8, E9, 10X, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1X9, 1EX, 20E, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2XX, 2EE, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39X, 3XE, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48X, 49E, 4E0, 501, 512, 514, 525, 536, 547, 558, 569, 57X, 58E, 5X0, 5E1.
The Friedman numbers up to 1000 are 121=11^{2}, 127=7×21, 135=5×31, 144=4×41, 163=3×61, 368=8^{6−3}, 376=6×73, 441=(4+1)^{4}, 445=5^{4}+4.
The Keith numbers up to 1000 are 11, 15, 1E, 22, 2X, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, XX, EE, 125, 215, 24X, 405, 42X, 654, 80X, 8X3, X59.
There are totally 71822 polydivisible numbers, the largest of which is 24-digit 606890346850EX6800E036206464. However, there are no E-digit polydivisible numbers contain the digits 1 to E exactly once each.
The candidate Lychrel numbers up to 1000 are 179, 1E9, 278, 2E8, 377, 3E7, 476, 4E6, 575, 5E5, 674, 6E4, 773, 7E3, 872, 8E2, 971, 9E1, X2E, X3E, X5E, X70, XXE, XE0, E2X, E3X, E5X, EXX. The only suspected Lychrel seed numbers up to 1000 are 179, 1E9, X3E and X5E. However, it is unknown whether any Lychrel number exists. (Lychrel numbers only known to exist in these bases: E, 15, 18, 22 and all powers of 2)
Most numbers that end with 2 are nontotient (in fact, all nontotients < 58 except 2X end with 2), except 2 itself, the first counterexample is 92, which equals φ(X1) = φ(E^{2}) and φ(182) = φ(2×E^{2}), next counterexample is 362, which equals φ(381) = φ(1E^{2}) and φ(742) = φ(2×1E^{2}), 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(p^{2}) = p(p−1), where p is a prime ends with E.
If we let the musical notes in an octave be numbers in the cyclic group Z_{10}: C=0, C#=1, D=2, Eb=3, E=4, F=5, F#=6, G=7, Ab=8, A=9, Bb=X, B=E (see pitch class and music scale) (thus, if we let the middle C be 0, then the notes in a piano are -33 to 40), then x and x+3 are small 3-degree, x and x+4 are big 3-degree, x and x+7 are perfect 5 degree (thus, we can use 7x for x = 0 to E to get the five degree cycle), etc. (since an octave is 10 semitones, a small 3-degree is 3 semitones, a big 3-degree is 4 semitones, and a perfect 5 degree is 7 semitones, etc.) (if we let an octave be 1, then a semitone will be 0.1, and we can write all 10 notes on a cycle, the difference of two connected notes is 26 degrees or $ \frac{\pi}{6} $ radians) Besides, the x major chord (x) is {x, x+4, x+7} in Z_{10}, and the x minor chord (xm) is {x, x+3, x+7} in Z_{10}, and the x major 7th chord (x^{M7}) is {x, x+4, x+7, x+E}, and the x minor 7th chord (x^{m7}) is {x, x+3, x+7, x+X}, and the x dominant 7th chord (x^{7}) is {x, x+4, x+7, x+X}, and the x diminished 7th triad (x^{dim7}) is {x, x+3, x+6, x+9}, since the frequency of x and x+6 is not simple integer fraction, they are not harmonic, and this diminished 7th triad is corresponding the beast number 666 (three 6's) (also the "beast tone" F#, which also corresponding the digit 6). Besides, x major scale uses the notes {x, x+2, x+4, x+5, x+7, x+9, x+E}, and x minor scale uses the notes {x, x+2, x+3, x+5, x+7, x+8, x+X}. Besides, the frequency of x+10 is twice as that of x, the frequency of x+7 is 1.6 (=3/2) times as that of x, and the frequency of x+5 is 1.4 (=4/3) times as that of x, they are all simple integer fractions (ratios of small integers), and they all have at most one digit after the duodecimal point, and we can found that 1.6^{10} = X9.8E5809 is very close to 2^{7} = X8, since 2^{17} = 2134X8 is very close to 3^{10} = 217669, the simple frequency fractions found for the scales are only 0.6, 0.8, 0.9, 1.4, 1.6 and 2, however, since the frequency of x+10 is twice as that of x, thus the frequency of x+1 (i.e. a semitone higher than x) is $ \sqrt[10]{2} $ (=2^{0.1}) times as that of x. Let f(x) be the frequency of x, then we have f(2)/f(0) = 9/8 (=1.16), f(4)/f(2) = X/9 (=1.14), and f(5)/f(4) = 14/13 (this number is very close to $ \sqrt[10]{2} $), and thus we have that f(5)/f(0) = (9/8) × (X/9) × (14/13) = 4/3. Also, we can found that 2^{0.5} is very close to 1.4, and 2^{0.7} is very close to 1.6.
All orders of non-cyclic simple group end with 0 (thus, all orders of unsolvable group end with 0), however, we can prove that no groups with order 10, 20, 30 or 40 are simple, thus 50 is the smallest order of non-cyclic simple group (thus, all groups with order < 50 are solvable), (50 is the order of the alternating group A_{5}, which is a non-cyclic simple group, and thus an unsolvable group) next three orders of non-cyclic simple group are 120, 260 and 360. (Edit: I found that this is not completely true (although this is true for all orders ≤ 14000), the smallest counterexample is 14X28, however, all such orders are divisible by 4 and either 3 or 5 (i.e. divisible by either 10 or 18), and all such orders have at least 3 distinct prime factors, by these conditions, the smallest possible such order is indeed 50 = 2^{2} × 3 × 5, next possible such order is 70 = 2^{2} × 3 × 7, however, by Sylow theorems, the number of Sylow 7-subgroups of all groups with order 70 (i.e. the number of subgroups with order 7 of all groups with order 70) is congruent to 1 mod 7 and divides 70, hence must be 1, thus the subgroup with order 7 is a normal subgroup of the group with order 70, thus all groups with order 70 have a nontrivial normal subgroup and cannot be simple groups)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E
appear in the repeating digits of 1/5 (exactly the even digits ≤ 5 (except 0 (the smallest digit)) and the odd digits ≥ 6 (except E (the largest digit)))
appear in the repeating digits of 1/7 (exactly the odd digits ≤ 5 and the even digits ≥ 6)
appear in the repeating digits of 1/11 (exactly the smallest digit (0) and the largest digit (E))
(note that all of 5, 7, and 11 are primes, and they are the only three primes ≤ 10+1 (=11) and divides neither 10 nor 10−1 (=E))