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:

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


#include <iostream>
#include <stdlib.h>
#include <fstream>
#include <strstream>
#include <list>
#include <set>
#include <math.h>
#include <iomanip>
#include <string>
#include <algorithm>
#include <iterator>
#include <gmp.h>
#include <gmpxx.h>
#include "form.h"

using namespace std;


int is_2_different_positive_2th_powers(mpz_class n)
{
  int fail = 1;
  for(mpz_class z = 1; fail && z * z < n; ++z){
    if(  mp_SquareQ(n - z * z )  )
    {
      mpz_class y =  mp_Sqrt(n - z * z );
      if ( y >= 1 && y != z   )
      { 
        fail = 0;
       // cout << n << " squares of: " << y << "  " << z << endl; 
      }
    }
  }
  return !fail;
}

int is_3_different_positive_3th_powers(mpz_class n)
{
  int fail = 1;
  for(mpz_class z = 2; fail && z * z * z < n; ++z){
  for(mpz_class y = 1; fail && y < z && y * y * y + z * z * z < n; ++y){
    mpz_class xxx = n - (y * y * y + z * z * z) ;
    if (mp_CubeQ( xxx) )
    {
       mpz_class x = mp_Cube_Root(xxx);
       if ( x > 0 && x != y && x != z)
       { 
        fail = 0;
       // cout << n << " cubes of: " << x << "  " << y << "  " << z << endl; 
      } 
    }

  }}
  return !fail;
}

int is_4_different_positive_4th_powers(mpz_class n)
{
  int fail = 1;
  for(mpz_class z = 3; fail && z * z * z * z < n; ++z){
  for(mpz_class y = 2; fail && y < z && y * y * y * y + z * z * z * z < n; ++y){
  for(mpz_class x = 1; fail && x < y && x * x * x * x + y * y * y * y + z * z * z * z < n; ++x){
     mpz_class wwww = n - (  x * x * x * x + y * y * y * y + z * z * z * z );
     if(  mp_Fourth_Power_Q(wwww ))
     {
       mpz_class w = mp_Fourth_Root(wwww);
       if( w > 0 && w != x && w != y && w != z)
      { 
        fail = 0;
       // cout << n << " fourth powers of: " <<  w << "  " << x << "  " << y << "  " << z << endl; 
      } 


     }
  }}}
  return !fail;
}



int is_5_different_positive_5th_powers(mpz_class n)
{
  int fail = 1;
  for(mpz_class z = 4; fail && to_power_5(z) < n; ++z){
  for(mpz_class y = 3; fail && y < z && to_power_5(y) + to_power_5(z) < n; ++y){
  for(mpz_class x = 2; fail && x < y && to_power_5(x) + to_power_5(y) + to_power_5(z) < n; ++x){
  for(mpz_class w = 1; fail && w < x &&  to_power_5(w) + to_power_5(x) + to_power_5(y) + to_power_5(z) < n; ++w){
     mpz_class vvvvv = n - (  to_power_5(w) + to_power_5(x) + to_power_5(y) + to_power_5(z) );
     if(  mp_Fifth_Power_Q(vvvvv))
     {
       mpz_class v =  mp_Fifth_Root(vvvvv);
       if( v > 0 && v != w && v != x && v != y && v != z) 
       { 
        fail = 0;
       // cout << n << " fifth powers of: " <<  v << "  " << w << "  " << x << "  " << y << "  " << z << endl; 
      } 


     }
  }}}}
  return !fail;
}


int is_6_different_positive_6th_powers(mpz_class n)
{
  int fail = 1;
  for(mpz_class z = 5; fail && to_power_6(z) < n; ++z){
  for(mpz_class y = 4; fail && y < z && to_power_6(y) + to_power_6(z) < n; ++y){
  for(mpz_class x = 3; fail && x < y && to_power_6(x) + to_power_6(y) + to_power_6(z) < n; ++x){
  for(mpz_class w = 2; fail && w < x &&  to_power_6(w) + to_power_6(x) + to_power_6(y) + to_power_6(z) < n; ++w){
  for(mpz_class v = 1; fail && v < w &&  to_power_6(v) + to_power_6(w) + to_power_6(x) + to_power_6(y) + to_power_6(z) < n; ++w){
     mpz_class uuuuuu = n - (  to_power_6(v) + to_power_6(w) + to_power_6(x) + to_power_6(y) + to_power_6(z)  );
     if(  mp_Sixth_Power_Q( uuuuuu))
     {
       mpz_class u =  mp_Sixth_Root(uuuuuu);
       if( u > 0 && u != v && u != w && u != x && u != y && u != z) 
       { 
        fail = 0;
     //   cout << n << " sixth powers of: " <<   u << "  " << v << "  " << w << "  " << x << "  " << y << "  " << z << endl; 
      } 


     }
  }}}}}
  return !fail;
}




int main()
{
   for(mpz_class p = 1; p <= 1000000000; p += 4 ){
   if( p % 10000 == 1 ) cerr << p << endl;
     if (  mp_PrimeQ(p) &&   is_2_different_positive_2th_powers(p) && is_3_different_positive_3th_powers(p) && is_4_different_positive_4th_powers(p)  && is_5_different_positive_5th_powers(p)  && is_6_different_positive_6th_powers(p)) cout << "    " << p << endl;
   }


    return 0 ;
}

