Given that each of A,B,C and D is a distinct positive integer.
Find the four smallest values of A+B+C+D such that each of A+B+C+D and A2+ B2+ C2+ D2 is a perfect square.
It looks as if every square from 5^2 onward is possible:
Sum Square
of Roots
A,B,C,D Sum Sqres
1 4 8 12 25 225 5 15
2 4 6 13 25 225 5 15
1 5 13 17 36 484 6 22
1 2 14 32 49 1225 7 35
1 10 16 22 49 841 7 29
2 4 13 30 49 1089 7 33
2 5 16 26 49 961 7 31
2 8 17 22 49 841 7 29
4 10 17 18 49 729 7 27
4 12 13 20 49 729 7 27
8 9 10 22 49 729 7 27
3 5 19 37 64 1764 8 42
3 9 25 27 64 1444 8 38
5 7 13 39 64 1764 8 42
5 7 23 29 64 1444 8 38
6 10 22 26 64 1296 8 36
7 9 15 33 64 1444 8 38
1 10 30 40 81 2601 9 51
1 20 28 32 81 2209 9 47
2 5 10 64 81 4225 9 65
2 6 20 53 81 3249 9 57
2 8 25 46 81 2809 9 53
2 8 33 38 81 2601 9 51
2 13 28 38 81 2401 9 49
4 5 22 50 81 3025 9 55
4 10 29 38 81 2401 9 49
4 14 29 34 81 2209 9 47
4 22 25 30 81 2025 9 45
5 6 8 62 81 3969 9 63
5 8 24 44 81 2601 9 51
5 10 14 52 81 3025 9 55
5 10 26 40 81 2401 9 49
5 16 22 38 81 2209 9 47
5 20 24 32 81 2025 9 45
6 17 26 32 81 2025 9 45
6 18 24 33 81 2025 9 45
8 10 26 37 81 2209 9 47
8 14 26 33 81 2025 9 45
8 16 17 40 81 2209 9 47
8 22 25 26 81 1849 9 43
10 12 25 34 81 2025 9 45
10 13 14 44 81 2401 9 49
10 17 26 28 81 1849 9 43
13 16 20 32 81 1849 9 43
2 10 14 74 100 5776 10 76
2 10 26 62 100 4624 10 68
2 18 34 46 100 3600 10 60
4 16 32 48 100 3600 10 60
5 13 21 61 100 4356 10 66
5 13 29 53 100 3844 10 62
5 17 37 41 100 3364 10 58
8 16 24 52 100 3600 10 60
9 13 33 45 100 3364 10 58
9 25 29 37 100 2916 10 54
13 17 33 37 100 2916 10 54
13 21 25 41 100 2916 10 54
1 2 30 88 121 8649 11 93
1 8 22 90 121 8649 11 93
1 18 44 58 121 5625 11 75
1 22 36 62 121 5625 11 75
2 8 53 58 121 6241 11 79
2 13 38 68 121 6241 11 79
2 14 25 80 121 7225 11 85
2 16 38 65 121 5929 11 77
2 16 46 57 121 5625 11 75
2 20 46 53 121 5329 11 73
2 25 26 68 121 5929 11 77
2 25 30 64 121 5625 11 75
2 28 38 53 121 5041 11 71
4 6 18 93 121 9025 11 95
4 6 33 78 121 7225 11 85
4 8 44 65 121 6241 11 79
4 9 10 98 121 9801 11 99
4 13 30 74 121 6561 11 81
4 13 34 70 121 6241 11 79
4 13 48 56 121 5625 11 75
4 22 30 65 121 5625 11 75
4 28 44 45 121 4761 11 69
4 29 40 48 121 4761 11 69
4 30 38 49 121 4761 11 69
4 33 34 50 121 4761 11 69
5 6 28 82 121 7569 11 87
5 6 40 70 121 6561 11 81
5 8 32 76 121 6889 11 83
5 8 52 56 121 5929 11 77
5 10 20 86 121 7921 11 89
5 16 32 68 121 5929 11 77
5 20 46 50 121 5041 11 71
5 26 28 62 121 5329 11 73
6 12 22 81 121 7225 11 85
6 12 25 78 121 6889 11 83
6 18 37 60 121 5329 11 73
6 25 40 50 121 4761 11 69
6 33 40 42 121 4489 11 67
8 9 46 58 121 5625 11 75
8 10 14 89 121 8281 11 91
8 13 36 64 121 5625 11 75
8 17 26 70 121 5929 11 77
8 25 34 54 121 4761 11 69
9 10 40 62 121 5625 11 75
9 24 28 60 121 5041 11 71
from
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For srtot = 2 To 100
tot = srtot * srtot
For a = 1 To tot / 4
For b = a + 1 To (tot - a) / 3
For c = b + 1 To (tot - a - b) / 2
d = tot - a - b - c
If d > c Then
t2 = a * a + b * b + c * c + d * d
sr = Int(Sqr(t2) + 0.5)
If sr * sr = t2 Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & Str(d) & " " & tot & Str(t2) & " " & srtot & Str(sr) & crlf
DoEvents
sct = sct + 1
If sct > 100 Then Exit Sub
End If
End If
Next
Next
Next
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
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Posted by Charlie
on 2015-07-25 13:22:55 |
computer-aided solution
