#6. Odd Distances

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Link to Slides · Link to recording

Prompts for discussion:

1. What fraction of the first $n$ Pythagorean Triplets (listed according to some appropriate order; sorry for the lazy phrasing) have all their side-lengths even? What other configurations (i.e, not a square) are valid answers to the question of “four points with all pairwise distances even”?

2. Question from @Vinay_V - can we get a configuration of four points among which five of the pairwise distances odd and one even? Preliminary step: what about just five pairwise odd distances?

3. Observe that $\det(2B) = 8\det(B)$, we have that $\det(2B) \equiv 0 \mod 8$, which contradicts the computation that $\det(2B) \equiv 4 \mod 8$. So I figure we could have “stopped short” here, and not needed the argument that involved comparing ranks.

4. Here’s the Wikipedia page on the Erdős–Anning theorem, and here’s the Geogebra playground.