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:

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


the smallest is
76913 squares of: 263  88
76913 cubes of: 40  17  20
76913 fourth powers of: 16  3  6  10
76913 fifth powers of: 9  1  2  4  7
76913

the second smallest is
1560402 squares of: 1239  159
1560402 cubes of: 101  45  76
1560402 fourth powers of: 35  5  12  14
1560402 fifth powers of: 17  1  6  8  10
1560402

Here is another one. It may or may not be the third smallest.
2091473 squares of: 1367  472
2091473 cubes of: 122  10  65
2091473 fourth powers of: 32  14  21  30
2091473 fifth powers of: 17  4  8  10  14
    2091473

