111 (number)

"Number 111" redirects here. For the 1919 film, see Number 111 (1919 film). For the 1938 film, see Number 111 (1938 film).

For other uses, see 111 (disambiguation).

← 110 111 112 →
← 110 111 112 113 114 115 116 117 118 119 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred eleven
Ordinal	111th (one hundred eleventh)
Factorization	3 × 37
Divisors	1, 3, 37, 111
Greek numeral	ΡΙΑ´
Roman numeral	CXI
Binary	11011112
Ternary	110103
Senary	3036
Octal	1578
Duodecimal	9312
Hexadecimal	6F16

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

In mathematics

111 is the fourth non-trivial nonagonal number, ^[1] and the seventh perfect totient number. ^[2]

111 is furthermore the ninth number such that its Euler totient ${\displaystyle \varphi (n)}$ of 72 is equal to the totient value of its sum-of-divisors:

${\displaystyle \varphi (111)=\varphi (\sigma (111)).}$ ^[3]

Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively). ^{[lower-alpha 1]}

Magic squares

111 is adjacent to 110 and 112, the minimal side lengths of perfect squared squares that are tiled by smaller squares of distinct side lengths.

The smallest magic square using only 1 and prime numbers has a magic constant of 111: ^[5]

31	73	7
13	37	61
67	1	43

Also, a six-by-six magic square using the numbers 1 to 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). ^{[lower-alpha 2]}

On the other hand, 111 lies between 110 and 112, which are the two smallest edge-lengths of squares that are tiled in the interior by smaller squares of distinct edge-lengths (see, squaring the square). ^[7]

Properties in certain radices

111 is ${\displaystyle R_{3}}$ or the second repunit in decimal, ^[8] a number like 11, 111, or 1111 that consists of repeated units, or ones. 111 equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are repdigits of the form ${\displaystyle 3n\times 37}$. 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 decimal, but is prime in base two, where 111₂ = 7₁₀. It is also prime in many other bases up to 128 (3, 5, 6, ..., 119) (sequence A002384 in the OEIS ). In base 10, it is furthermore a strobogrammatic number, ^[9] as well as a Harshad number. ^[10]

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

Nelson

Main article: Nelson (cricket)

In 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".

Particularly in cricket, multiples of 111 are called a double Nelson (222), triple Nelson (333), quadruple Nelson (444; also known as a salamander) and so on.

A score of 111 is considered by some to be unlucky. To combat the supposed bad luck, some watching lift their feet off the ground. Since an umpire cannot sit down and raise his feet, the international umpire David Shepherd had a whole retinue of peculiar mannerisms if the score was ever a Nelson multiple. He would hop, shuffle, or jiggle, particularly if the number of wickets also matched—111/1, 222/2 etc.

In other fields

111 is also:

• The atomic number of the element roentgenium (Rg).
• The chemical compound 1,1,1-trichloroethane is a chlorinated hydrocarbon that was used as an industrial solvent with a trade name "Solvent 111".
• The emergency telephone number in New Zealand; see 111 (emergency telephone number).
• NHS 111, a medical helpline in England and Scotland.
• It is the lowest positive integer requiring seven syllables to name in English (British and Commonwealth), or six syllables (by dropping the "and") in American English.
• Occasionally it is referred to as "eleventy-one", as read in The Fellowship of the Ring by J.R.R. Tolkien. ^[11]
• The first Pacific (4-6-2) locomotive in Great Britain was the Great Western Railway's No. 111 The Great Bear of 1908, designed by George Jackson Churchward. This locomotive was rebuilt into a 4-6-0 Castle Class engine in 1924, renamed Viscount Churchill, and retained the fleet number 111 until scrapped in July 1953.
• In Pakistani history, Pakistan Army's 111 Brigade is notorious for being used for staging the coups to overthrow the elected governments in order to impose martial law.

See also

• List of highways numbered 111

Notes

1. Also, ^[3]
   • The 111st composite number 146 ^[4] is the twelfth number whose totient value is the same value held by its sum-of-divisors. The sequence of nonagonal numbers that precede 111 is {0, 1, 9, 24, 46, 75}, ^[1] members which add to 146 (without including 9).
   • 357, in turn the index of 444 as a composite, ^[4] is the twentieth such number, following 333.
   • The composite index of 1000 is 831, ^[4] the thirty-fifth member in this sequence of numbers to have a totient also shared by its sum-of-divisors, where 1000 is 1 + 999.
   The only two numbers in decimal less than 1000 whose prime factorisations feature primes concatenated into a new prime are 138 and 777 (as 2 × 3 × 23 and 3 × 7 × 37, respectively), which add to 915. This sum represents the 38th member in the aforementioned sequence. ^[3]
2. Relatedly, 111 is also the magic constant of the n-Queens Problem for n = 6. ^[6]

References

1. Sloane, N. J. A. (ed.). "SequenceA001106(9-gonal (or enneagonal or nonagonal) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 26 May 2016.
2. Sloane, N. J. A. (ed.). "SequenceA082897(Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 26 May 2016.
3. Sloane, N. J. A. (ed.). "SequenceA006872(Numbers k such that phi(k) is phi(sigma(k)).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 3 February 2024.
4. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 3 February 2024.
5. Henry E. Dudeney (1917). Amusements in Mathematics (PDF). London: Thomas Nelson & Sons, Ltd. p. 125. ISBN   978-1153585316. OCLC   645667320.
6. Sloane, N. J. A. (ed.). "SequenceA006003(a(n) = n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
7. Gambini, Ian (1999). "A method for cutting squares into distinct squares". Discrete Applied Mathematics . 98 (1–2). Amsterdam: Elsevier: 65–80. doi: 10.1016/S0166-218X(99)00158-4 . MR   1723687. Zbl   0935.05024.
8. Sloane, N. J. A. (ed.). "SequenceA002275(Repunits: (10^n - 1)/9. Often denoted by R_n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Sloane, N. J. A. (ed.). "SequenceA000787(Strobogrammatic numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 7 May 2022.
10. Sloane, N. J. A. (ed.). "SequenceA005349(Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 26 May 2016.
11. 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.

Further reading

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134