<s>[INST] Minimize $a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$ with distinct positive integers 
Find the minimum value of the following:
  $$a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$$
  where all numbers are different/distinct positive integers.

I know the answer (see below), but want to confirm the same. 
Is there any way to prove following conjecture?

Conjecture. There is always unique way to write down $\sum_{i=1}^{n} a_{i}^n$ for any arbitrary value of $n$ such that it gives same value for all values of $n$.


Answer is given below, spoiler alert:

 $$1^5+2^5+4^5+7^5+9^5 = 3^4+6^4+10^4+16^4 = 17^3+20^3+40^3 = 88^2 + 263^2 = 76913$$

 [/INST] Thanks Will Jagy for sharing the code.
I had used following VBA code in excel 

Sub Power5()
For u = 1 To 10
    For v = u + 1 To 10
        For w = v + 1 To 10
            For x = w + 1 To 10
                For y = x + 1 To 10
                    mysum = u ^ 5 + v ^ 5 + w ^ 5 + x ^ 5 + y ^ 5
                    For p = 1 To 100
                        For q = p + 1 To 100
                            For r = q + 1 To 100
                                For s = r + 1 To 100
                                    If p ^ 4 + q ^ 4 + r ^ 4 + s ^ 4 = mysum Then
                                        For i = 1 To 100
                                            For j = i + 1 To 100
                                                For k = j + 1 To 100
                                                    If i ^ 3 + j ^ 3 + k ^ 3 = mysum Then
                                                        For a = 1 To 1000
                                                            For b = a + 1 To 1000
                                                                If a ^ 2 + b ^ 2 = mysum Then
                                                                    n = n + 1: Cells(n, 1) = u & "| " & v & "| " & w & "| " & x & "| " & y & "<|> " & p & "| " & q & "| " & r & "| " & s & "<|> " & i & "| " & j & "| " & k & "<|> " & a & "| " & b
                                                                    GoTo Mynext
                                                                End If
                                                            Next
                                                        Next
                                                    End If
                                                Next
                                            Next
                                        Next
                                    End If
                                Next
                            Next
                        Next
                    Next
                Next
            Next
        Next
Mynext:
    Next
Next
End Sub

</s>