Studies in Two-Factor Authentication: Solution

By Brandon Avila

Answer:
BOTNET

As clued by the flavor text, this puzzle has to do with factoring primes in the complex plane. The image is a 250x250 section of the first quadrant in the complex plane beginning at (0, 0) in the bottom left corner. Each black dot represents a Gaussian prime (a prime number in the ring of Gaussian integers), and each colored dot represents a composite number with exactly two factors.

For each color (red through purple), the composite numbers are factored, and their factors are connected pairwise in the plane by a line as follows:

`B`

: Red

43 + 80*i* = (8 + 3*i*)(8 + 7*i*)

76 + 65*i* = (8 + 3*i*)(11 + 4*i*)

68 + 87*i* = (8 + 5*i*)(11 + 4*i*)

58 + 103*i* = (8 + 5*i*)(11 + 6*i*)

46 + 125*i* = (8 + 7*i*)(11 + 6*i*)

`O`

: Orange

6 + 3*i* = (2 + *i*)(3)

12 + 3*i* = (4 + *i*)(3)

10 + 11*i* = (4 + *i*)(3 + 2*i*)

4 + 7*i* = (2 + *i*)(3 + 2*i*)

`T`

: Yellow

47 + 140*i* = (8 + 7*i*)(12 + 7*i*)

79 + 100*i* = (10 + 7*i*)(10 + 3*i*)

`N`

: Green

11 + 24*i* = (4 + *i*)(4 + 5*i*)

19 + 34*i* = (6 + *i*)(4 + 5*i*)

31 + 36*i* = (6 + *i*)(6 + 5*i*)

`E`

: Blue

15 + 104*i* = (6 + 5*i*)(10 + 9*i*)

33 + 58*i* = (6 + 5*i*)(8 + 3*i*)

57 + 178*i* = (10 + 9*i*)(12 + 7*i*)

60 + 109*i* = (11 + 4*i*)(8 + 7*i*)

`T`

: Purple

1 + 12*i* = (1 + 2*i*)(5 + 2*i*)

9 + 6*i* = (3 + 2*i*)(3)

Taken in order, these letters spell out the solution, ** BOTNET**.