<s>[INST] To show that there are $3$ numbers between $n^2$ and $(n+1) ^2$ such that they satisfy a certain property. Here is the problem

Show that for any natural number $n$, one can find three distinct
natural numbers $a, b, c$ between $n^2$ and $(n + 1)^2$ such that $a^2 + b^2$ is divisible
by $c$.

It seems like here are several ways to express $a^2 + b^2$.
\begin{array}
 aa^2 + b^2 &= (a - b) ^2 +  2ab \\
 &= (a + b) ^2 - 2ab \\
 &= \frac { (a + b) ^2 + (a - b)^2 }{2} \\
 &= 2b^2 + (a + b)(a - b)
\end{array}
What the problem is literally saying: Find algebraic expressions $X,Y$ and $Z$ such that $n^2 \lt X,Y,Z \lt(n + 1)^2$ and $Y^2 + Z^2 = AX$, for some expression $A$. (Hence, $X|Y^2 + Z^2$)
In the original problem, $X$ is $c$, $Y$ and $Z$ are $b$ and $a$.
People with enough experience might know the answer at once, but is it possible to actually find the expressions from identities/relations? For example the relations that I have found above?
From my trying it is very difficult because one needs to search for expression that are restricted between $n^2$ and $(n + 1)^2$, and then the factoring. What I tried was just plugging random expressions to $Y$ and $Z$ and trying to find a suitable $X$ that suffices the factoring, but to no avail.
Please help, and thanks for that!
 [/INST] There is in fact another way to do this:

Thm 1: Let $N$ be a sufficiently large integer. Then between $N$ and
$N +\lceil 2\sqrt{N} \rceil$ there are 3 integers $A$, $B$, $C$ such that
$C|(A^2+B^2)$.

To prove Thm 1, we make the following claim:

Claim 2: For each $M$ and $k$, the equation $(M-k)^2 \equiv_k (M+k)^2 \equiv_M k^2$.

We now finish the proof of Thm 1: So let $k_1,k_2$ be distinct positive integers both no larger than $\sqrt{N}$ such that $N+\sqrt{N} > k_1^2+k^2_2 > N$, picking
$k_1=\lfloor \sqrt{N} \rfloor$ and $k_2 = O(N^{1/4})$ should work.
Then let $C=k^2_1+k^2_2$, and then let $A=C+k_1$ and $B=C+k_2$. Then $A,B,C$ satisfy the conditions of Thm 1. $\surd$
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