I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$)
Question:
Can we find all sequences of non negative integers ($\forall i\in \mathbb{N} \,\, a_i\in \mathbb{N}$) such that $$\begin{align} \forall i,j,k,l\in\mathbb{N} && i^2+j^2=k^2+l^2 &\Rightarrow a_i^2+a_j^2=a_k^2+a_l^2\end{align}$$
My try: I know the Euler's complete solution to the Diophantine equation :
$$x^2+y^2=z^2+t^2$$ $(ab+cd,ac-bd,ab-cd,ac+bd)$ but this is not very useful.
I also tried another way which is if we suppose that the sequence is injective ($a_i\neq a_j$ if $i\neq j$, maybe later we can prove that either the sequence is constant or this will be true), and under this assumption we can deduce that if $a_i+a_j$ is odd then: $$ d(|a_i^2-a_j^2|)\geq |i^2-j^2| \\ r_2(a_i^2+a_j^2)\geq r_2(i^2+j^2)$$
but the problem is we don't know much about the function $d(n)$ the number of divisors of $n$ and $r_2(n)$ the number of ways to write $n$ as the sum of two squares.
Expected solution: The only sequences I found are $a_i=c$ or $a_i=ki$ But is it the only sequences that fits our needs?, can we find a sequence with two values $c$ and $c'$ that fits the assertion?
Any comments, suggestions, any consequences of such assertion are welcome,Thank you for you help.