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 Equal Power Sum Sets (Posted on 2017-10-11)
41+51+121=21+91+101
42+52+122=22+92+102

Are equal power sum sets (EPSS) for second power, but the sum 4+5+12 is not the smallest. Find the smallest with all distinct positive integers.

Find EPSS of four numbers each (8 distinct positive integers) where the first, second and third powers have equal sums.

Keep going, if possible, to find higher order EPSS where fourth, fifth powers, etc. have equal sums.

Note: The highest EPSS I have is fifth power.

 No Solution Yet Submitted by Jer No Rating

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 computer solution Comment 1 of 1
Part 1:

1+5+6 = 2+3+7 and
1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2

Other pairs of sets are shown (those using no number higher than 12):

Duplicates occur as no check was done for reversing LHS and RHS.

1 5 6     2 3 7
1 6 8     2 4 9
1 7 10     2 5 11
1 8 9     3 4 11
2 3 7     1 5 6
2 4 9     1 6 8
2 5 11     1 7 10
2 6 7     3 4 8
2 7 9     3 5 10
2 8 11     3 6 12
2 9 10     4 5 12
3 4 8     2 6 7
3 4 11     1 8 9
3 5 10     2 7 9
3 6 12     2 8 11
3 7 8     4 5 9
3 8 10     4 6 11
4 5 9     3 7 8
4 5 12     2 9 10
4 6 11     3 8 10
4 8 9     5 6 10
4 9 11     5 7 12
5 6 10     4 8 9
5 7 12     4 9 11
5 9 10     6 7 11
6 7 11     5 9 10
6 10 11     7 8 12
7 8 12     6 10 11

DefDbl A-Z
Dim crlf\$, used(12)

Form1.Visible = True

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

For a = 1 To 12
used(a) = 1
For b = a + 1 To 12
used(b) = 1
For c = b + 1 To 12
used(c) = 1
For x = 1 To 12
If used(x) = 0 Then
used(x) = 1
For y = x + 1 To 12
If used(y) = 0 Then
used(y) = 1
For z = y + 1 To 12
If used(z) = 0 Then
used(z) = 1

If a + b + c = x + y + z Then
If a * a + b * b + c * c = x * x + y * y + z * z Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & "    " & Str(x) & Str(y) & Str(z) & crlf
End If
End If

used(z) = 0
End If
DoEvents
Next z
used(y) = 0
End If
Next y
used(x) = 0
End If
Next x
used(c) = 0
Next c
used(b) = 0
Next b
used(a) = 0
Next a

Text1.Text = Text1.Text & crlf & " done"

End Sub

Part 2:

The below is, again, really only one solution with left and right sides of the equation reversed. The first row in each case shows the first power; the second row, the second power; and the third row, the third power.

1 5 8 12     2 3 10 11
1 25 64 144      4 9 100 121
1 125 512 1728        8 27 1000 1331

2 3 10 11     1 5 8 12
4 9 100 121      1 25 64 144
8 27 1000 1331        1 125 512 1728

DefDbl A-Z
Dim crlf\$, used(12)

Form1.Visible = True

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

For a = 1 To 12
used(a) = 1
For b = a + 1 To 12
used(b) = 1
For c = b + 1 To 12
used(c) = 1
For d = c + 1 To 12
used(d) = 1
For x = 1 To 12
If used(x) = 0 Then
used(x) = 1
For y = x + 1 To 12
If used(y) = 0 Then
used(y) = 1
For z = y + 1 To 12
If used(z) = 0 Then
used(z) = 1
For w = z + 1 To 12
If used(w) = 0 Then
used(w) = 1

If a + b + c + d = x + y + z + w Then
If a * a + b * b + c * c + d * d = x * x + y * y + z * z + w * w Then
If a * a * a + b * b * b + c * c * c + d * d * d = x * x * x + y * y * y + z * z * z + w * w * w Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & Str(d) & "    " & Str(x) & Str(y) & Str(z) & Str(w) & crlf
Text1.Text = Text1.Text & a ^ 2 & Str(b ^ 2) & Str(c ^ 2) & Str(d ^ 2) & "     " & Str(x ^ 2) & Str(y ^ 2) & Str(z ^ 2) & Str(w ^ 2) & crlf
Text1.Text = Text1.Text & a ^ 3 & Str(b ^ 3) & Str(c ^ 3) & Str(d ^ 3) & "       " & Str(x ^ 3) & Str(y ^ 3) & Str(z ^ 3) & Str(w ^ 3) & crlf
Text1.Text = Text1.Text & crlf
End If
End If
End If

used(w) = 0
End If
DoEvents
Next w
used(z) = 0
End If
Next z
used(y) = 0
End If
Next y
used(x) = 0
End If
Next x
used(d) = 0
Next d
used(c) = 0
Next c
used(b) = 0
Next b
used(a) = 0
Next a

Text1.Text = Text1.Text & crlf & " done"

End Sub

 Posted by Charlie on 2017-10-11 15:13:30

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