Curious properties of 11, 111, 1111, …

Older post:   Series of Num3ers 1111…
 
 
11   is a prime number.
111 = 3 * 37
1111 = 11 * 101
11111 = 41 * 271
…..
1111111111111111111   (19 digits)   is a prime number.
11111111111111111111111   (23 digits)   is a prime number.
 
 
Curious properties of 11:

(1)

1^2 = 1
11^2 = 121     1+2+1 = 4 = 2^2
111^2 = 12321     1+2+3+2+1 = 9 = 3^2
1111^2 = 1234321     1+2+3+4+3+2+1 = 16 = 4^2
11111^2 = 123454321     1+2+3+4+5+4+3+2+1 = 25 = 5^2
111111^2 = 12345654321     1+2+3+4+5+6+5+4+3+2+1 = 36 = 6^2
1111111^2 = 1234567654321     1+2+3+4+5+6+7+6+5+4+3+2+1 = 49 = 7^2
11111111^2 = 123456787654321     1+2+3+4+5+6+7+8+7+6+5+4+3+2+1 = 64 = 8^2

111111111^2 = 12345678987654321    
1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1 = 81 = 9^2

1111111111^2 = 1234567900987654321    
1+2+3+4+5+6+7+9+0+0+9+8+7+6+5+4+3+2+1 = 82 = 1^2 + 9^2

11111111111^2 = 123456790120987654321
1+2+3+4+5+6+7+9+0+1+2+0+9+8+7+6+5+4+3+2+1 = 85 = 2^2 + 9^2 = 6^2 + 7^2

111111111111^2 = 12345679012320987654321
1+2+3+4+5+6+7+9+0+1+2+3+2+0+9+8+7+6+5+4+3+2+1 = 90 = 3^2 + 9^2

1111111111111^2 = 1234567901234320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+3+2+0+9+8+7+6+5+4+3+2+1 = 97 = 4^2 + 9^2

11111111111111^2 = 123456790123454320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+4+3+2+0+9+8+7+6+5+4+3+2+1 = 106
106 = 5^2 + 9^2

111111111111111^2 = 12345679012345654320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+6+5+4+3+2+0+9+8+7+6+5+4+3+2+1 = 117
117 = 6^2 + 9^2

1111111111111111^2 = 1234567901234567654320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+6+7+6+5+4+3+2+0+9+8+7+6+5+4+3+2+1 = 130
130 = 3^2 + 11^2 = 7^2 + 9^2

11111111111111111^2 = 123456790123456787654320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+6+7+8+7+6+5+4+3+2+0+9+8+7+6+5+4+3+2+1 = 145
145 = 1^2 + 12^2 = 8^2 + 9^2

111111111111111111^2 = 12345679012345678987654320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+0+9+8+7+6+5+4+3+2+1 = 162
162 = 9^2 + 9^2

1111111111111111111^2 = 1234567901234567900987654320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+6+7+9+0+0+9+8+7+6+5+4+3+2+0+9+8+7+6+5+4+3+2+1= 163
163 = 1^2 + 9^2 + 9^2

11111111111111111111^2 = 123456790123456790120987654320987654321
1+2+3+4+5+6+7+9+0+1+2+3+4+5+6+7+9+0+1+2+0+9+8+7+6+5+4+3+2+0+9+8+7+6+5+4+3+2+1 = 166
166 = 2^2 + 9^2 + 9^2

The pattern is:
1^2,   2^2,   3^2,   4^2,   5^2,   6^2,   7^2,   8^2,   9^2
then
1^2 + 9^2,   2^2 + 9^2,   3^2 + 9^2, ….,   9^2 + 9^2
next,
1^2 + 9^2 + 9^2,   2^2 + 9^2 + 9^2, ….,   9^2 + 9^2 + 9^2
 
 
(2)

11^2   =   121   =   (22 * 22)/(1 + 2 + 1)
111^2   =   12321   =   (333 * 333)/(1 + 2 + 3 + 2 + 1)
1111^2   =   1234321   =   (4444 * 4444)/(1 + 2 + 3 + 4 + 3 + 2 + 1)

and so on.
 
 
(3)

(1 + 1)/(1 + 1) = 1
(11 + 1)/(1 + 1) = 6
(111 + 1)/(1 + 1) = 56
(1111 + 1)/(1 + 1) = 556
(11111 + 1)/(1 + 1) = 5556
(11…11 + 1)/(1 + 1) = 5…56

(111 + 11)/(1 + 1) = 61
(1111 + 11)/(1 + 1) = 561
(11111 + 11)/(1 + 1) = 5561
(111…11 + 11)/(1 + 1) = 5…561

(1111 + 111)/(1 + 1) = 611
(11111 + 111)/(1 + 1) = 5611
(111111 + 111)/(1 + 1) = 55611
(111…111 + 111)/(1 + 1) = 5…5611

and so on.
 
 
(4)

3 * 37 = 111
33 * 3367 = 111111
333 * 333667 = 111111111
3333 * 33336667 = 111111111111
33333 * 3333366667 = 111111111111111
333333 * 333333666667 = 111111111111111111

and so on.
 
 
(5)

(0 * 9)   +   1   =   1
(1 * 9)   +   2   =   11
(12 * 9)   +   3   =   111
(123 * 9)   +   4   =   1111
(1234 * 9)   +   5   =   11111
(12345 * 9)   +   6   =   111111
(123456 * 9)   +   7   =   1111111
(1234567 * 9)   +   8   =   11111111
(12345678 * 9)   +   9   =   111111111
(123456789 * 9)   +   10   =   1111111111
(1234567900 * 9)   +   11   =   11111111111
(12345679011 * 9)   +   12   =   111111111111

and so on.
 
 
(6)

11   =   1^2   +   1^2   +   3^2

1111   =   11^2   +   12^2   +   13^2   +   14^2   +   15^2   +   16^2
 
 
 

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About benvitalis

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

2 Responses to Curious properties of 11, 111, 1111, …

  1. Pingback: Curious properties of 11, 111, 1111, … | Benvitalis's Blog

  2. Pingback: List of Num3ers – Archives (Jan. 2012 – Oct. 2012) | Fun With Num3ers

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