Please note that the solution must not require more equations to solve as do the variables increase.
Apparently, $(a+b+c\cdots)\neq(a^{2}+b^{2}+c^{2}\cdots)$ seems pretty obviously to be true given that all variables $a,b,c\cdots$ are distinct natural numbers, there is no disproof I can think of (also please note that the number of variables on the RHS and LHS are same and equal in quantity, i.e. $(a+b) \neq (a^{2} + b^{2})$ or $(a+b+c+d) \neq (a^{2}+b^{2}+c^{2}+d^{2}$).
If this is indeed true, it becomes obvious that there is only one unique solution for all variables that satisfies both equations (only and only two are given, as the example towards the end of this post). Wolfram|Alpha does the job to a certain degree, except it doesn't recognize that all variables must have different values, and also counts permutations of values among the variables as possible solution.
So, how do I find a solution given two equations, as in;
$a+b+c+d+e=1837, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=1234567$
BTW the solution is $a=66, b=79, c=169, d=628, e=895$, also, if the conjecture is false, are there any counterexamples (again, please remember that all variables are distinct).
Thanks to whomsoever has offered to help.