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)
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