In Mathematics
111 is a perfect totient number.
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.
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 100: 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, and 99. 111 is also prime in base 111 (= 1243310).
In radix 18 the number 111 is the third power of seven 73 (= 34310) which is the only radix where 111 is a 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:
| 24 | 16 | 33 | 23 | 10 | 5 |
| 11 | 15 | 28 | 8 | 13 | 36 |
| 20 | 14 | 2 | 31 | 25 | 19 |
| 1 | 18 | 6 | 29 | 27 | 30 |
| 21 | 22 | 7 | 17 | 32 | 12 |
| 34 | 26 | 35 | 3 | 4 | 9 |
(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. It is also an nonagonal number.
In base 10, it is a Harshad number.
Read more about this topic: 111 (number)
Famous quotes containing the word mathematics:
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)
“Why does man freeze to death trying to reach the North Pole? Why does man drive himself to suffer the steam and heat of the Amazon? Why does he stagger his mind with the mathematics of the sky? Once the question mark has arisen in the human brain the answer must be found, if it takes a hundred years. A thousand years.”
—Walter Reisch (1903–1963)