So of course I had to write a program to find the cubes.

If A = B^3 + C^3, then either B^3 or C^3 must be no more than A/2. Assume B < C. Then B^3 <= A/2 so B <= cuberoot(A/2).

When you enter a number and click the Go button, the program loops variable B from 0 to cuberoot(A/2). (In Visual Basic, you can calculate the cube root or a number by taking the log of the number, dividing by 3, and then taking the exponent of the result.)

For each possible value B, the code calculates A - B^3 and takes the integer cube root to see what C would be. If C^3 exactly equals A - B^3, then we have a solution.

The program continues for other values of B to see if there are other solutions.

Private Sub btnGo_Click(ByVal sender As System.Object, _
    ByVal e As System.EventArgs) Handles btnGo.Click
    txtResults.Text = ""
    Me.Cursor = Cursors.WaitCursor
    Application.DoEvents()

    Dim A As Decimal = Decimal.Parse(txtNumber.Text)
    Dim max_B As Decimal = CDec(Exp(Log(A / 2) / 3))
    If max_B * max_B * max_B > A Then max_B -= 1

    ' Try numbers up to max_B.
    For B As Decimal = 0 To max_B
        Dim remainder As Decimal = A - (B * B * B)
        Dim C As Decimal = CLng(Exp(Log(remainder) / 3))
        If C * C * C = remainder Then
            txtResults.Text = txtResults.Text & _
                B & "^3 + " & _
                C & "^3 = " & _
                (B * B * B) & " + " & (C * C * C) & vbCrLf
            txtResults.Select(0, 0)
            Application.DoEvents()
        End If
    Next B

    Beep()
    Me.Cursor = Cursors.Default
End Sub

Much to my surprise, Bender's serial number is not expressible as a sum of two positive cubes. However, Bender's serial number 2716057 = (-951) ^ 3 + 952 ^ 3 = -860085351 + 862801408.

If you allow negative numbers, it seems as if anything might be possible.

Notes:

• The smallest number expressible as the sum of N cubes is called Taxicab(N).
• Taxicab(2) = 1729 is the smallest number expressible in two different ways as the sum of two (positive) cubes 1^3 + 12^3 and 9^3 + 10^3
• Taxicab(3) = 87539319
• Taxicab(4) = 6963472309248
• Taxicab(5) = 48988659276962496
• It is not known whether Taxicab(6) = 24153319581254312065344, but that is a candidate (it is expressible in 6 ways as a sum of two cubes)