x^3 + y^3 + z^3 = w^3.
The smallest set of positive integers x,y,z,w qualifying as a solution of the bolded equation above is:
3,4,5,6
a.Find some other sets of distinct integers, not necessarily positive.
b. Provide some comments on "The sum of 3 cubes is a cube" equation.
(In reply to
computer lists (spoilers for part a) by Charlie)
Allowing negatives with absolute values in ascending order (total of absolute values of x, y and z <=200):
3 4 5 6
-3 -4 6 5
-3 -5 6 4
1 6 8 9
-4 -5 6 3
-1 -6 9 8
-1 -8 9 6
-1 9 10 12
1 -9 12 10
1 -10 12 9
-6 -8 9 1
6 8 10 12
-2 9 15 16
-6 -8 12 10
2 -9 16 15
-6 -10 12 8
2 12 16 18
-8 -10 12 6
3 10 18 19
9 10 -12 1
-2 -12 18 16
-3 -10 19 18
2 -15 16 9
-2 -16 18 12
9 12 15 18
7 14 17 20
-9 -12 18 15
-2 18 20 24
-3 -18 19 10
9 15 -16 2
-7 -14 20 17
-9 -15 18 12
4 17 22 25
2 -18 24 20
-7 -17 20 14
3 18 24 27
-12 -15 18 9
2 -20 24 18
-4 -17 25 22
-12 -16 18 2
-10 -18 19 3
-3 -18 27 24
12 16 20 24
-2 15 33 34
2 -15 34 33
-4 -22 25 17
-14 -17 20 7
-4 18 30 32
-12 -16 24 20
-10 19 24 27
11 15 27 29
-3 -24 27 18
4 -18 32 30
-11 -15 29 27
10 -19 27 24
-12 -20 24 16
-9 16 33 34
18 19 21 28
2 17 40 41
9 -16 34 33
-2 -17 41 40
-3 27 30 36
4 24 32 36
15 20 25 30
-16 -20 24 12
10 -24 27 19
6 20 36 38
18 20 -24 2
-4 -24 36 32
-6 -20 38 36
-17 -22 25 4
-15 -20 30 25
-18 -19 28 21
3 -27 36 30
4 -30 32 18
-11 -27 29 15
-18 -21 28 19
-19 -21 28 18
2 -33 34 15
3 -30 36 27
-18 -24 27 3
-15 -25 30 20
19 24 -27 10
6 32 33 41
-15 -27 29 11
-4 -32 36 24
18 24 30 36
5 30 40 45
-20 -25 30 15
3 36 37 46
9 -33 34 16
-12 31 33 40
14 28 34 40
-6 27 45 48
-18 -24 36 30
-6 -32 41 33
-17 26 36 39
-4 36 40 48
-5 -30 45 40
-6 -33 41 32
-6 -36 38 20
16 23 41 44
18 30 -32 4
6 -27 48 45
-14 -28 40 34
15 33 -34 2
17 -26 39 36
-2 -40 41 17
12 -31 40 33
-16 -23 44 41
16 33 -34 9
-3 22 59 60
12 19 53 54
-18 -30 36 24
21 28 35 42
3 -22 60 59
-3 -36 46 37
-12 -19 54 53
12 -33 40 31
-3 -37 46 36
8 34 44 50
-8 29 50 53
4 -36 48 40
-9 22 57 58
-14 -34 40 28
9 -22 58 57
-5 -40 45 30
6 36 48 54
8 -29 53 50
-24 -30 36 18
-21 -28 42 35
4 -40 48 36
-8 -34 50 44
17 -36 39 26
-24 -32 36 4
9 30 54 57
-12 38 43 51
27 30 -36 3
-20 -36 38 6
27 30 37 46
-17 24 54 55
-6 -36 54 48
-9 -30 57 54
17 -24 55 54
24 32 40 48
-17 -40 41 2
-21 -35 42 28
6 -45 48 27
-4 30 66 68
-5 45 50 60
12 -38 51 43
-16 -41 44 23
26 36 -39 17
4 -30 68 66
-8 -44 50 34
-28 -34 40 14
-27 -30 46 37
-8 36 60 64
-24 -32 48 40
31 33 -40 12
7 42 56 63
-28 -35 42 21
12 -43 51 38
15 42 49 58
-20 38 48 54
22 30 54 58
-32 -33 41 6
14 23 70 71
29 34 44 53
-6 -48 54 36
8 -36 64 60
-14 -23 71 70
-23 -41 44 16
27 36 45 54
5 -45 60 50
-22 -30 58 54
-27 -37 46 30
8 -50 53 29
-7 -42 63 56
20 -38 54 48
-24 -40 48 32
-30 -37 46 27
21 42 51 60
5 -50 60 45
-15 -42 58 49
-30 -40 45 5
-18 32 66 68
-29 -34 53 44
36 38 42 56
-27 -36 54 45
4 34 80 82
7 54 57 70
18 -32 68 66
-12 -53 54 19
-36 -37 46 3
-4 -34 82 80
-6 54 60 72
8 48 64 72
-9 -54 57 30
27 45 -48 6
30 40 50 60
-32 -40 48 24
3 -59 60 22
-5 48 69 76
-15 -49 58 42
20 -48 54 38
-21 -42 60 51
9 -57 58 22
12 40 72 76
36 40 -48 4
-7 -56 63 42
17 -54 55 24
-19 -53 54 12
-27 -45 54 36
-29 -44 53 34
22 51 54 67
-8 -48 72 64
-12 -40 76 72
-17 38 73 76
-34 -44 50 8
-2 41 86 89
5 -48 76 69
12 51 66 75
-10 45 75 80
-30 -40 60 50
-36 -38 56 42
-7 -54 70 57
17 -38 76 73
-34 -44 53 29
2 -41 89 86
6 -54 72 60
8 -60 64 36
-21 -51 60 42
29 50 -53 8
33 44 55 66
38 43 -51 12
-11 30 92 93
24 54 -55 17
-7 -57 70 54
11 -30 93 92
-22 -54 58 30
-36 -42 56 38
9 54 72 81
10 -45 80 75
36 38 61 69
-36 -45 54 27
31 33 72 76
-38 -42 56 36
22 57 -58 9
4 -66 68 30
6 -60 72 54
-12 -51 75 66
34 39 65 72
-36 -48 54 6
-30 51 58 67
-7 63 70 84
-15 54 71 80
-22 -51 67 54
-30 -50 60 40
-31 -33 76 72
38 48 -54 20
22 59 -60 3
-30 -54 57 9
12 64 66 82
25 31 86 88
-30 -54 58 22
17 40 86 89
-22 -54 67 51
-33 -44 66 55
-36 -38 69 61
-8 -64 72 48
-9 -54 81 72
-25 -31 88 86
36 48 60 72
12 31 102 103
-34 -39 72 65
-12 -31 103 102
-17 -40 89 86
25 48 74 81
38 43 66 75
19 60 69 82
21 43 84 88
30 -51 67 58
15 -54 80 71
-20 33 96 97
-42 -49 58 15
5 -69 76 48
-6 45 99 102
10 60 80 90
20 -33 97 96
25 38 87 90
-40 -50 60 30
3 34 114 115
-3 -34 115 114
6 72 74 92
18 -66 68 32
-21 -43 88 84
-24 62 66 80
28 56 68 80
6 -45 102 99
-12 -66 75 51
20 54 79 87
-25 -38 90 87
-42 -51 60 21
7 -63 84 70
-25 -48 81 74
-33 -55 66 44
-14 -70 71 23
15 50 90 95
30 -58 67 51
45 50 -60 5
-12 54 90 96
28 53 75 84
-36 -48 72 60
-38 -43 75 66
39 52 65 78
-12 -64 82 66
-34 52 72 78
-1 64 94 103
26 55 78 87
-30 57 72 81
33 45 81 87
-42 56 61 69
-8 72 80 96
-10 -60 90 80
-12 -66 82 64
-12 -72 76 40
-15 -50 95 90
32 46 82 88
36 60 -64 8
7 -70 84 63
-19 -60 82 69
-20 -54 87 79
-42 -56 63 7
-9 -72 81 54
12 -54 96 90
19 53 90 96
-23 -70 71 14
-28 -56 80 68
30 66 -68 4
34 -52 78 72
10 -75 80 45
11 66 88 99
-28 -53 84 75
-33 -45 87 81
38 48 79 87
-44 -55 66 33
-4 -80 82 34
15 -71 80 54
17 -73 76 38
24 -62 80 66
-32 -46 88 82
32 66 -68 18
-36 -61 69 38
42 -56 69 61
1 -64 103 94
-6 44 118 120
13 51 104 108
-19 -53 96 90
24 38 106 108
-26 -55 87 78
30 -57 81 72
-36 -60 72 48
-38 -61 69 36
42 56 70 84
-39 -52 78 65
6 -44 120 118
-6 -72 92 74
-19 -69 82 60
-23 63 84 94
-24 -38 108 106
24 -66 80 62
16 47 108 111
32 54 85 93
-34 -65 72 39
-4 67 101 110
-6 -74 92 72
-13 -51 108 104
16 68 88 100
42 -61 69 56
-51 -54 67 22
-38 -48 87 79
-16 -47 111 108
-16 58 100 106
-27 48 99 102
54 57 63 84
50 61 64 85
8 -72 96 80
-11 -66 99 88
-18 44 114 116
-24 63 89 98
-28 -68 80 56
-39 -65 72 34
51 58 -67 30
2 -86 89 41
6 51 120 123
27 -48 102 99
18 -44 116 114
-23 60 95 102
-31 -72 76 33
-32 -54 93 85
-38 -66 75 43
-6 -51 123 120
9 55 116 120
-9 81 90 108
-10 -80 90 60
12 72 96 108
16 -58 106 100
23 -63 94 84
-25 -74 81 48
45 60 75 90
-48 -60 72 36
4 -67 110 101
-29 60 92 99
-33 -72 76 31
-54 -57 70 7
-14 63 105 112
-39 -65 78 52
-42 -56 84 70
30 -72 81 57
8 -80 96 72
-9 -55 120 116
-16 -68 100 88
34 -72 78 52
-43 -66 75 38
-48 -64 72 8
23 -60 102 95
24 -63 98 89
15 82 89 108
18 60 108 114
-20 -79 87 54
-24 76 86 102
-35 59 92 98
54 60 -72 6
56 61 -69 42
58 59 69 90
-28 -75 84 53
38 73 -76 17
29 -60 99 92
-40 -72 76 12
54 60 74 92
14 -63 112 105
-34 48 108 110
35 70 85 100
-51 64 75 82
-26 -78 87 55
-12 -72 108 96
-17 -86 89 40
-18 -60 114 108
34 -48 110 108
35 -59 98 92
48 64 80 96
-51 -66 75 12
-21 -84 88 43
45 69 79 97
48 69 -76 5
13 78 104 117
33 70 92 105
-45 -60 90 75
-52 -65 78 39
-54 -57 84 63
-4 41 151 152
11 -92 93 30
-34 -80 82 4
-42 -70 84 56
-50 -61 85 64
4 -41 152 151
51 -64 82 75
1 -94 103 64
9 -81 108 90
-11 -88 99 66
12 -90 96 54
13 65 121 127
-25 -86 88 31
-50 -64 85 61
-8 60 132 136
-10 90 100 120
-15 -90 95 50
29 75 96 110
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr(13) + Chr(10)
For tot = 2 To 200
For x = 0 To tot / 3
x3 = x * x * x
For y = x + 1 To (tot - x) / 2
z = tot - x - y
If z > y Then
y3 = y * y * y: z3 = z * z * z
For xs = -1 To 1 Step 2
For ys = -1 To 1 Step 2
For zs = -1 To 1 Step 2
'If xs * x <> ys * y And ys * y <> zs * z And xs * x <> zs * z Then
w3 = xs * x3 + ys * y3 + zs * z3
If w3 >= 0 Then
w = Int(w3 ^ (1 / 3) + 0.5)
If w * w * w = w3 Then
Text1.Text = Text1.Text & mform(xs * x, "#####") & mform(ys * y, "#####") & mform(zs * z, "#####") & " " & mform(w, "#####") & crlf
End If
End If
'End If
DoEvents
Next
Next
Next
End If
Next
Next
Next tot
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
|
|
Posted by Charlie
on 2018-06-21 14:44:30 |
re: computer lists (spoilers for part a) -- continued, with mixed signs
