DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
p3 = 3
For base3 = 2 To 31
p3 = p3 * 3
k = 1
Do
k = k + 1
n1 = 1
t1 = n1
didOne = 0
Do
n1 = n1 + 1
t1 = Int(n1 ^ k + 0.5)
If t1 > p3 Then Exit Do
didOne = 1
t2 = p3 - t1
n2 = Int(t2 ^ (1 / k) + 0.5)
If t2 = Int(n2 ^ k + 0.5) And t2 > 1 Then
' If gcd(n1, n2) = 1 Then
Text1.Text = Text1.Text & base3 & Str(p3) & " = " & n1 & "^" & k & "+" & n2 & "^" & k
Text1.Text = Text1.Text & " " & gcd(n1, n2) & crlf
ct = ct + 1
' End If
End If
DoEvents
Loop
Loop Until didOne = 0
Next
Text1.Text = Text1.Text & crlf & ct & " done"
End Sub
Function gcd(a, b)
x = a: y = b
Do
q = Int(x / y)
z = x - q * y
x = y: y = z
Loop Until z = 0
gcd = x
End Function
was originally written to check that the two integers were coprime, but no solutions were found, so that condition was commented out. The results now show that all the solutions have two numbers that are multiples of 3, and the powers (k) are all 3. The gcd's of the two numbers are all powers of 3.
power
of 3 gcds
3^5 = 243 = 3^3+6^3 3
3^5 = 243 = 6^3+3^3 3
3^8 = 6561 = 9^3+18^3 9
3^8 = 6561 = 18^3+9^3 9
3^11 = 177147 = 27^3+54^3 27
3^11 = 177147 = 54^3+27^3 27
3^14 = 4782969 = 81^3+162^3 81
3^14 = 4782969 = 162^3+81^3 81
3^17 = 129140163 = 243^3+486^3 243
3^17 = 129140163 = 486^3+243^3 243
3^20 = 3486784401 = 729^3+1458^3 729
3^20 = 3486784401 = 1458^3+729^3 729
3^23 = 94143178827 = 2187^3+4374^3 2187
3^23 = 94143178827 = 4374^3+2187^3 2187
3^26 = 2541865828329 = 6561^3+13122^3 6561
3^26 = 2541865828329 = 13122^3+6561^3 6561
3^29 = 68630377364883 = 19683^3+39366^3 19683
3^29 = 68630377364883 = 39366^3+19683^3 19683
Edited on August 25, 2016, 2:12 pm
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Posted by Charlie
on 2016-08-25 14:11:11 |
Computer exploration
