Are there any integer solution to $\sum{x_i} = \sum{y_i}$ and $\sum{x_i}^2 = \sum{y_i}^2$

Question

I have two integer sets ($X=\{x_1,x_2,...,x_k\}, Y=\{y_1,y_2,...,y_k\}$) with equal sizes ($k$) with unique values. I was just wondering if there are any integer solutions to: $$\sum_i x_i = \sum_i y_i$$ and $$\sum_i {x_i}^2 = \sum_i {y_i}^2$$ for any $k>2$?

There is an answer for $k=2$ here: https://www.quora.com/Does-a-b-c-d-and-ab-cd-mean-a-b-c-d

but I'm just wondering can we generalize it for $k>2$?

@univalence I think when OP says “unique values” they mean unique across both sets, so $X=Y$ is forbidden. — Erick Wong, Apr 02 '21 at 20:35
@player3236, Thanks for sharing the link. I understood it's kind of related to "Multigrade Equation", https://mathworld.wolfram.com/MultigradeEquation.html — Mahdi Hajiabadi, Apr 02 '21 at 20:45
https://artofproblemsolving.com/community/c3046h1057888_one_system_of_diophantine_equations — individ, Apr 06 '21 at 11:19
https://artofproblemsolving.com/community/c3046h1053878_the_system_of_equations_quotbquot — individ, Apr 06 '21 at 11:20
https://artofproblemsolving.com/community/c3046h1053831_the_system_of_equations_quotaquot — individ, Apr 06 '21 at 11:21

score 2 · Answer 1 · answered Apr 02 '21 at 20:20

2

Maybe $(1,2,-3)$ and $(-1,-2,3)$ can be a counterexample?

answered Apr 02 '21 at 20:20

Hypernova

782

This is a much nicer base case than the one in my answer. Anyone who wants a positive solution can just add $4$ to everything, so it seems likely to be the minimal solution in some sense. – Erick Wong Apr 02 '21 at 21:25

score 1 · Accepted Answer · answered Apr 02 '21 at 20:29

Yes. One of the comments links to a similar question which gives $\{ 1,6,8 \}$ and $\{ 2,4,9\}$ as examples (and there are many more for small values of $k$).

One of the beautiful things about this problem is that it is intrinsically translation-invariant (also scale-invariant): we actually have, as polynomials,

$$(x+1y)^2 + (x+6y)^2 + (x+8y)^2 = (x+2y)^2 + (x+4y)^2 + (x+9y)^2.$$

So we can use a single example for $k=3$ or $k=4$, etc. to generate any number of examples. By choosing these examples well, we can glue multiple shifted copies together and keep the entries unique. So this should easily build examples for all $k>2$.

Are there any integer solution to $\sum{x_i} = \sum{y_i}$ and $\sum{x_i}^2 = \sum{y_i}^2$

2 Answers2