Curious properties of 11, 111, 1111, …

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Fun With Num3ers

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

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E.A. Bennet: Mrs. Jung died at 10.30 this morning. C.G. came up to the Niehus’s house to tell me and to say goodbye.

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Küsnacht, 23rd November 1955Arrived at Zürich station at 12.25 and was met by C.G. He told me Mrs. Jung was very ill and so I would not be staying with them but with his daughter Marianne Niehus. B…

Source: E.A. Bennet: Mrs. Jung died at 10.30 this morning. C.G. came up to the Niehus’s house to tell me and to say goodbye.