Let natural numbers $a, b > 2$ be adjacent if $|a^2 - b^2|$ is a square number. One can find a 3-clique.
For example 153, 185, 697. The questions are: does there exist a 4-clique? Is this graph connected?



4-cliques of pythagorean triples graph and its connectivity
Seems that here is a proof that the graph is connected


Let $C$ be a component.


1. $C$ contains an even number, as an odd number $2k+1$ is adjacent to $2k(k+1)$.
2. $C$ contains a number divisible by $6$: if $2k\in C$ with $3\nmid k$, then $k^2-1\in C$ as well.
3. If $3k\in C$ and $p$ is a prime, then $3kp\in C$. Induction on $p$. The base cases $p=2,3$ follow from adjacencies $3k-5k-13k-12k-15k-17k-8k-10k-6k$ and $\dots-15k-9k$. For the step, use the hypothesis to see that $C$ contains $3k(p^2-1)$, then $6kp$, and then $3kp$ by the base case.
4. The graph is connected. Indeed, any two components contain some numbers $3k$ and $3n$, and then both contain $3kn$.


