111 (number)

"Number 111" redirects here. For the 1919 film, see Number 111 (1919 film). For the 1938 film, see Number 111 (1938 film).
← 110 111 112 →
Cardinal one hundred eleven
Ordinal 111th
(one hundred and eleventh)
Factorization 3 × 37
Divisors 1, 3, 37, 111
Roman numeral CXI
Binary 11011112
Ternary 110103
Quaternary 12334
Quinary 4215
Senary 3036
Octal 1578
Duodecimal 9312
Hexadecimal 6F16
Vigesimal 5B20
Base 36 3336

111 (One hundred [and] eleven) is the natural number following 110 and preceding 112.

In mathematics

111 is a perfect totient number.[1]

111 is R3 or the second repunit, a number like 11, 111, or 1111 that consists of repeated units, or 1's. It equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are of the form 3n × 37. As a repunit, it also follows that 111 is a palindromic number.

All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 1112 = 710. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119 (sequence A002384 in the OEIS)

In base 18, the number 111 is 73 (= 34310) which is the only base where 111 is a perfect power.

The smallest magic square using only 1 and prime numbers has a magic constant of 111:

31 73 7
13 37 61
67 1 43

A six-by-six magic square using the numbers 1 through 36 also has a magic constant of 111:

1 11 31 29 19 20
2 22 24 25 8 30
3 33 26 23 17 9
34 27 10 12 21 7
35 14 15 16 18 13
36 4 5 6 28 32

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).

111 is also the magic constant of the n-Queens Problem for n = 6.[2] It is also an nonagonal number.[3]

In base 10, it is a Harshad number.[4]

Nelson

Main article: Nelson (cricket)

The number 111 is sometimes called "a Nelson" after Admiral Nelson, who allegedly only had "One Eye, One Arm, One Leg" near the end of his life. This is in fact inaccurate - Nelson never lost a leg. Alternate meanings include "One Eye, One Arm, One Ambition" and "One Eye, One Arm, One Arsehole".

It is particularly known as a score in cricket. A score of 111 or multiples thereof (222 = "double nelson", 333 = "triple nelson" and so on) is considered unlucky by some in English cricket: most famously by the international umpire David Shepherd, who had a whole retinue of peculiar mannerisms - hops, shuffles, jiggles and so on - that he would indulge in if the score was ever a "Nelson" multiple. Particularly if the number of wickets also matched - 111/1, 222/2 etc.

In other fields

111 is also:

See also

References

  1. ↑ "Sloane's A082897 : Perfect totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
  2. ↑ "Sloane's A006003". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ↑ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
  4. ↑ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
  5. ↑ John Ronald Reuel Tolkien (1993). The fellowship of the ring: being the first part of The lord of the rings. HarperCollins. ISBN 978-0-261-10235-4. Retrieved 29 August 2011.

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