111 number: Encyclopedia II - 111 number - In mathematics

111 number - In mathematics

One hundred eleven is R₃ or the second repunit, a number like 11, 111, or 1111 that consists of repeated units, or 1's. It equals 3 x 37, therefore all triplets (numbers like 222 or 666) in base ten are of the form 3n x 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 111 = 7 in base ten. It is also prime in these other bases 3 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).

The smallest magic square using only prime numbers (and 1) 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).

This number is also the magic constant of n-Queens Problem for n = 6, an enneagonal number and a Harshad number.

111 is the lowest positive integer requiring six syllables to name in American English, or seven syllables (including "and") in British English.

Adapted from the Wikipedia article "In mathematics", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki