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 Unity Root Count (Posted on 2015-03-09)
Consider sets A = {u: u18 = 1} and B = {v: v48 = 1}, where neither of the sets A and B can contain any purely real element.
Define C = {u*v: u ε A, v ε B}.

Not counting any purely real element, determine the total number of distinct complex elements in C.

 No Solution Yet Submitted by K Sengupta No Rating

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 solution | Comment 1 of 2
Set A has 18 - 2 = 16 elements, while B has 48 - 2 = 46 elements.

The roots are all located on a unit circle in the Argand plane. The LCM of  18 and 48 is 144, so all products will be at multiples of 1/144 of a full circle (adding angles to do multiplication). Elements of A are multiples of 8/144 of a circle, and of B are multiples of 3/144 of a circle.

This program does the addition of the 144ths and computes the resulting 144ths. 0 and 72 mod 144 correspond to real values and are not used as multiplicands (addends in the additions) or products (sums in the additions).

DefDbl A-Z
Dim member(143), crlf\$

Text1.Text = ""
crlf\$ = Chr(13) + Chr(10)
Form1.Visible = True

' count 1/144's of a circle in the Argand plane
For a = 8 To 144 - 8 Step 8
If a * 2 <> 144 Then
For b = 3 To 144 - 3 Step 3
If b * 2 <> 144 Then
prodAngle = (a + b) Mod 144
If prodAngle <> 0 And prodAngle <> 72 Then
member(prodAngle) = member(prodAngle) + 1
End If
End If
Next
End If
Next a

For m = 1 To 143
If member(m) > 0 Then
ct = ct + 1
Text1.Text = Text1.Text & m & Str(member(m)) & crlf
End If
Next
Text1.Text = Text1.Text & ct & crlf

Text1.Text = Text1.Text & "done"
DoEvents

End Sub

Listed first are counts of the various possible multiples of 1/144 of a circle in the products angles on the Argand plane:

1 6
2 6
3 4
4 6
5 6
6 4
7 6
8 4
9 4
10 6
11 6
12 4
13 6
14 6
15 4
16 4
17 6
18 4
19 6
20 6
21 4
22 6
23 6
24 2
25 6
26 6
27 4
28 6
29 6
30 4
31 6
32 4
33 4
34 6
35 6
36 4
37 6
38 6
39 4
40 4
41 6
42 4
43 6
44 6
45 4
46 6
47 6
48 2
49 6
50 6
51 4
52 6
53 6
54 4
55 6
56 4
57 4
58 6
59 6
60 4
61 6
62 6
63 4
64 4
65 6
66 4
67 6
68 6
69 4
70 6
71 6
73 6
74 6
75 4
76 6
77 6
78 4
79 6
80 4
81 4
82 6
83 6
84 4
85 6
86 6
87 4
88 4
89 6
90 4
91 6
92 6
93 4
94 6
95 6
96 2
97 6
98 6
99 4
100 6
101 6
102 4
103 6
104 4
105 4
106 6
107 6
108 4
109 6
110 6
111 4
112 4
113 6
114 4
115 6
116 6
117 4
118 6
119 6
120 2
121 6
122 6
123 4
124 6
125 6
126 4
127 6
128 4
129 4
130 6
131 6
132 4
133 6
134 6
135 4
136 4
137 6
138 4
139 6
140 6
141 4
142 6
143 6

In total 142 possibilities are represented. Only the two real values are discounted from all 144 possibilities.

Of course, two of the products are purely imaginary also. Do they count as complex?

 Posted by Charlie on 2015-03-09 17:39:29

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