One hundred and eleven

In mathematics, 111, or one hundred (and) eleven in base ten, 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 through 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).