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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 n2is 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 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 322 = X04 = 1444X)

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, 8EE8, 9119, X44X and E77E, but none of them are squares)

The palindromic squares up to 10 digits are: (not including 0) (see OEIS: A029738)

n

n-digit palindromic squares (0 is not included)

square roots

number of n-digit palindromic squares (0 is not included)

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.

Rn2 (where Rn 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, there 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 X (in fact, no perfect powers end with 2, 6 or X), 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 nk+2 ends with the same digit as nk, thus, if i≥2, j≥2 and i and j have the same parity, then ni and nj 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 x2x=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)

The trimorphic numbers up to 10000 are 0, 1, 3, 4, 5, 7, 8, 9, E, 15, 47, 53, 54, 5E, 61, 68, 69, 75, X7, E3, EE, 115, 253, 368, 369, 4X7, 5EE, 601, 715, 853, 854, 969, XX7, EEE, 14X7, 2369, 3853, 3854, 4715, 5EEE, 6001, 74X7, 8368, 8369, 9853, X715, EEEE.

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.

Unit digit of nk

k

n

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)

Final two digits of nk

k

n

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 10n−1 (= λ(10n)).

Digital root of nk

k

n

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 104.

In the following table, Fn is the n-th Fibonacci number, and Ln 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

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 F2X begins with L1X, and F2E begins with L1E)

If n ends with 0, then Fn ends with 00, i.e. if n is divisible by 10, then Fn is divisible by 100.

The final two digits of Fn+20 is the same as the final two digits of Fn, the cycle of the final two digits of Fn is {00, 01, 01, 02, 03, 05, 08, 11, 19, 2X, 47, 75, 00, 75, 75, 2X, X3, 11, E4, 05, E9, 02, EE, 01}. Note that not only the unit digit, but also the dozens digit of a Fibonacci number cannot be 6.

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 2×10n−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 OEIS: A066853).

Except 0 = F0 and 1 = F1 = F2, the only square Fibonacci number is 100 = F10 (100 is indeed 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

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 221 = E2X20X8. For all digits 1≤d≤E, there exists 0≤n≤21 such that 2n starts with the digit d.

21XE = 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 272 = E8196855X752550E455X1594 = 77371252455336267181195264X, it has only 22 digits in base X)

The number 229 = 2368 (see power of 2#Powers of two whose exponents are powers of two) is very close to googol (10100), since it has EE digits. (thus, the Fermat number F9 (=229+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 (= 13 + 103) = 509 + 6E4 (= 93 + X3) (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).

All values of n (greater than 45) for incrementally largest values of minimal x > 1 (or minimal y > 0) satisfying Pell's equation {\displaystyle x^{2}-ny^{2}=1} end with 1, and their dozens digit are odd.

If n ends with 2 and n/2 is prime (or 1), then the denominator of the Bernoulli number {\displaystyle B_{n}} is 6 (this is also true for some (but not all) n ends with X and n/2 is prime). (if the denominator of the Bernoulli number {\displaystyle B_{n}} is 6, then n ends with 2 or X, but n/2 needs not to be prime or 1, the first counterexample is n = 82, the denominator of the Bernoulli number {\displaystyle B_{82}} is 6, but 82/2 = 41 = 72 is neither prime nor 1)

{\displaystyle {\sqrt {2}}} is very close to 1.5, since a near-value for {\displaystyle {\sqrt {2}}} is 15/10 (=N4/P4, where Nn is nth NSW number, and Pn is nth Pell number, Nn/Pn is very close to {\displaystyle {\sqrt {2}}} when n is large). Besides, {\displaystyle {\sqrt {5}}} is very close to 2.2X, since a near-value for {\displaystyle {\sqrt {5}}} is 22X/100 (=L10/F10, where Ln is nth Lucas number, and Fn is nth Fibonacci number, Ln/Fn is very close to {\displaystyle {\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.

Regular n-gon is constructible using neusis, or an angle trisector if and only if the reciprocal of {\displaystyle \varphi (n)} is terminating number (where {\displaystyle \varphi } is Euler's totient function), thus the n ≤ 200 such that regular n-gon is constructible using neusis, or an angle trisector are 3, 4, 5, 6, 7, 8, 9, X, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 2X, 2E, 30, 31, 32, 33, 34, 36, 39, 40, 43, 44, 46, 48, 49, 50, 53, 54, 55, 58, 5X, 60, 61, 62, 64, 66, 68, 69, 70, 71, 76, 77, 7E, 80, 81, 86, 88, 89, 90, 91, 93, 94, 96, 99, 9E, X0, X6, X8, XX, E1, E3, E4, E8, 100, 102, 104, 108, 109, 110, 114, 116, 117, 120, 122, 123, 130, 132, 135, 139, 13X, 140, 141, 142, 143, 150, 154, 156, 160, 162, 163, 165, 166, 168, 170, 176, 17X, 180, 183, 187, 190, 193, 194, 195, 197, 198, 1X2, 1X6, 1X8, 1X9, 1E4, 1E9, 200.

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 mod 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 {\displaystyle {\tbinom {n}{k}}={\tfrac {n!}{k!(n-k)!}}} is squarefree for all 0 ≤ k ≤ n, i.e. all numbers in the nth row of the Pascal's triangle are squarefree (the topmost row (i.e. the row which contains only one 1) of the Pascal's triangle is the 0th row, not the 1st row).

By Benford's law, the probability for the leading digit d (d ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}) occurs (for some sequences, e.g. powers of 2 (1, 2, 4, 8, 14, 28, 54, X8, ...) and Fibonacci sequence (1, 1, 2, 3, 5, 8, 11, 19, ...)) are:

d

probability

d

probability

1

34.2%

7

7.9%

2

1E.6%

8

6.X%

3

14.8%

9

6.1%

4

10.E%

X

5.6%

5

X.7%

E

5.1%

6

8.E%

(Note: the percentage in the list are also in duodecimal, i.e. 20% means 0.2 or {\displaystyle {\frac {20}{100}}={\frac {1}{6}}}, 36% means 0.36 or {\displaystyle {\frac {36}{100}}={\frac {7}{20}}}, 58.7% means 0.587 or {\displaystyle {\frac {587}{1000}}})

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 frequency of primes ending with 1 is relatively low, but the frequency of primes ending with 5, 7 and E are nearly equal. (since all squares of primes except 4 and 9 end with 1, no squares of primes end with 5, 7 or E)

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

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.

If p is Sophie Germain prime end with E, then 2p-1 cannot be prime since it must be divisible by 2p+1 (this is not true for all Sophie Germain primes, 75 is Sophie Germain prime, but 275-1 is prime, and this is also not true for all primes end with E, 8E ends with E, but 28E-1 is prime). (see Mersenne prime)

A prime p is Gaussian prime (prime in the ring {\displaystyle Z[i]}, where {\displaystyle i={\sqrt {-1}}}) with no imaginary part if and only if p ends with 7 or E (or p=3).

A prime p is Eisenstein prime (prime in the ring {\displaystyle Z[\omega ]}, where {\displaystyle \omega ={\frac {-1+{\sqrt {3}}i}{2}}}) with no imaginary part if and only if p ends with 5 or E (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).

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. {\displaystyle \left({\frac {10}{p}}\right)=-1}, where {\displaystyle \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 = {\displaystyle \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 = {\displaystyle \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 {\displaystyle {\frac {a}{p}}=0.{\overline {a_{1}a_{2}a_{3}...a_{n}}}}, then ai + ai+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 1060 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/pis 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 61X = 51, 131X = XE, 181X = 131, 461X = 325, 491X = 34E, see OEIS: A073761) (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 7X = 7, 23X = 1E, 47X = 3E, 59X = 4E, 167X = 11E, see OEIS: A000353) (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 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

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., {\displaystyle {\frac {1}{17}}} has period level 3, thus the numbers {\displaystyle {\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, {\displaystyle {\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 {\displaystyle {\frac {a}{15}}} with integer 1 ≤ a ≤ 14 from the cycle 08579214E36429X7.

All Fermat primes except 3 are full-reptend primes. (Mersenne primes 23−1, 25−1, 27−1, 215−1, 217−1, 275−1 and 28E−1 are also full-reptend primes, but 211−1, 227−1, 251−1 and 2X7−1 are not)

There are only 9 repunit primes up to R1000R2R3R5R17R81R91R225R255 and R4X5 (Rn is the repunit with length n). If p is a Sophie Germain prime other than 2, 3 and 5, then Rp is divisible by 2p+1, thus Rp 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 10c-1 = 1 mod c) which are not divisible by E, they are called "deceptive primes", if n is deceptive prime, then Rn 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 using only 0's and 1's.

Smallest multiple of n using 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 = R4 = 1111 = 5×11×25, besides, this k for n = 46E is 2X3E, which makes k×n = R7 = 1111111, a repunit semiprime, and this k for n = 19E is a X8-digit prime number, with k×n = RXE, 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 Rn 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 10c = 10 mod c) which are not divisible by E) Besides, the deceptive primes are exactly the repunit pseudoprimes which are coprime to 10)

A Fermat number {\displaystyle F_{n}} is prime if and only if {\displaystyle F_{n}} is prime if and only if {\displaystyle 10^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}. (see Pépin's test).

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 ndivisible 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.

The smallest n-parasitic numbers for n = 1 to n = E are (leading zeros are not allowed)

n

Smallest n-parasitic number

Digits

Period of

1

1

1

1/E

2

10631694842

E

2/1E

3

2497

4

7/2E = 1/5

4

10309236X88206164719544

1E

4/3E

5

1025355X9433073X458409919E715

25

5/4E

6

1020408142854X997732650X18346916306

2E

6/5E

7

101899E864406E33XX15423913745949305255E17

35

7/6E

8

131X8X

6

X/7E = 2/17

9

101419648634459E9384E26E533040547216X1155E3E12978X399

45

9/8E

X

14E36429X7085792

14

12/9E = 2/15

E

1011235930336X53909X873E325819E9975055E54X3145X42694157078404491E

55

E/XE

Write the recurring digits of 1/45 (=0.2872E3X23205525X784640XX4E9349081989E6696143757E117, 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{1n} = 34×10n+(10n−1)/E = (309×10n−1)/E and it can be factored to ((19×10n/2−1)/E) × (19×10n/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{1E78}, 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{1n} 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 10×582781E5+1, it has 459655 digits.

All numbers of the form 1{5}1 are composite (proof: 1{5n}1 = (14×10n+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 1010100 are 15, 57, 5E, 117, 11E, 5EEE (the smallest representative prime of the permutation set).

The non-repdigit circular primes below 1010100 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 below 1010 are 1685 and 5E685, note that both of these two 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 = 0100010001002 = 44414 is a repdigit in base 14, and 2046 = 1101101101102 = 66668 is also a repdigit in base 8, see Wieferich prime#Binary periodicity of p − 1)

n

Wieferich prime base n below 106

n

Wieferich prime base n below 106

1

all primes

11

2, 5EE, 703407

2

771, 2047

12

25, 255

3

E, 406217

13

14X37

4

771, 2047

14

771, 2047

5

2, 1002E, 1E51E

15

2, 3, 22771, 243XE

6

32355, 21962E

16

5, 7, 31, 237, 1776E, 51E0E7

7

5, 1E854E

17

3, 7, 11, 37, E5

8

3, 771, 2047

18

1E5, 22X75

9

2, E, 406217

19

2

X

3, 347

1X

11, 481, 64E605

E

5E

1E

11, 9E8251

10

1685, 5E685

20

5, 12X01

The primeval numbers are 1, 2, 13, 15, 57, 115, 117, 125, 135, 157, 1017, 1057, 1157, 1257, 125E, 157E, 167E, ...

There are 1, 2, 3, 5 and 6-digit (but not 4-digit) narcissistic numbers, there are totally 73 narcissistic numbers (not including 0), 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 OEIS: A161949)

The only two factorions are 1 and 2.

The only 7 Dudeney numbers are 1, 5439, 61E4, 705E, 16E68, 18969, 1X8E4.

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 1000X (=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)

There are no happy primes below 10000, the happy primes below 100000 are 11031, 1233E, 13011, 1332E, 16377, 17367, 17637, 22E8E, 2331E, 233E1, 23955, 25935, 25X8E, 28X5E, 28XE5, 2X8E5, 2E82E, 2E8X5, 31011, 31101, 3123E, 3132E, 31677, 33E21, 35295, 35567, 35765, 35925, 36557, 37167, 37671, 39525, 4878E, 4X7X7, 53567, 55367, 55637, 56357, 57635, 58XX5, 5X82E, 5XX85, 606EE, 63575, 63771, 66E0E, 67317, 67371, 67535, 6E60E, 71367, 71637, 73167, 76137, 7XX47, 82XE5, 82EX5, 8487E, 848E7, 84E87, 8874E, 8X1X7, 8X25E, 8X2E5, 8X5X5, 8XX17, 8XX71, 8E2X5, 8E847, 92355, 93255, 93525, 95235, X1X87, X258E, X285E, X2E85, X85X5, X8X17, XX477, XX585, E228E, E606E, E822E, EX825.

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 153X (=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 + X3, 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 cycles 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, the harshad numbers up to 100 are exactly the union of the numbers ≤ 10, the multiples of E and the multiples of 10 up to 100 (except EE). 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.

There are only 5 sum-product numbers, namely 0, 1, 128, 173, 353.

The Rhonda numbers up to 10000 are 3X8, 568, 2389, 2689, 27E6, 29E4, 4297, 4974, 5483, 6X35, 6E64, 7662, 86E8, 8864, 94E4, X867, XE36. Besides, 3X8 is the smallest number that is a Rhonda number to some base (base 10).

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 only self-descriptive number is 821000001000.

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, 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 10-digit polydivisible numbers contain the digits 0 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(E2) and φ(182) = φ(2×E2), next counterexample is 362, which equals φ(381) = φ(1E2) and φ(742) = φ(2×1E2), 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 E.

If we only have the numbers end with 1, 5 or 9 (i.e. numbers = 1 mod 4), then the unique factorization theorem is not true: 309 = 9 × 41 = 192 and all of 9, 41 and 19 are primes (since they are not product of any two smaller numbers = 1 mod 4 and > 1) (see Hilbert number). Similarly, if we only have the numbers end with 1, 2, 7 or X (i.e. numbers = 1 mod 3), then the unique factorization theorem is not true: 84 = 4 × 21 = X2 and all of 4, 21 and X are primes (since they are not product of any two smaller numbers = 1 mod 3 and > 1)

If we let the musical notes in an octave be numbers in the cyclic group Z10: 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) (x and y are the same pitch class if and only if the unit digit of x and the unit digit of y are the same) (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 minor third, x and x+4 are major third, x and x+7 are perfect fifth (thus, we can use 7x for x = 0 to E to get the circle of fifths), etc. (since an octave is 10 semitones, a minor third is 3 semitones, a major third is 4 semitones, and a perfect fifth 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 {\displaystyle {\frac {\pi }{6}}} radians, i.e. {\displaystyle {\frac {1}{10}}} of a cycle) Besides, the x major chord (x) is {xx+4, x+7} in Z10, and the x minor chord (xm) is {xx+3, x+7} in Z10, and the x major 7th chord (xM7) is {xx+4, x+7, x+E}, and the x minor 7th chord (xm7) is {xx+3, x+7, x+X}, and the x dominant 7th chord (x7) is {xx+4, x+7, x+X}, and the x diminished 7th triad (xdim7) is {xx+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 related to the beast number 666 (three 6's). Besides, x major scale uses the notes {xx+2, x+4, x+5, x+7, x+9, x+E}, and x minor scale uses the notes {xx+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 1.610 = X9.8E5809 is very close to 27 = X8, since 217 = 2134X8 is very close to 310 = 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 {\displaystyle {\sqrt[{10}]{2}}} (=20.1) 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 {\displaystyle {\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 20.5 is very close to 1.4, and 20.7 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 A5, 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 = 22 × 3 × 5, next possible such order is 70 = 22 × 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)