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Triangle Radicals

Posted 3 years ago
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I'm expanding a discussion from Integer Triangle Radicals conjecture here. At the end, there's code for an eventual Demonstration, Triangle Radicals.

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the triangle. For the first 24 squarefree numbers, here are the smallest triangles with a given radical.

smallest triangles of a given radical

Now consider sequence $G$.
$$G = (24,8,1,0,8,8,3,0,8,24,1,0)$$

Integer Triangle Radicals conjecture:
For a triangle with radical $r$, let $j = r \mod{12}$.
Then value $4 A/\sqrt{r}$ is divisible by $G_j$ (with $j \in \{1,...,12\})$.

Stated another way, $G_j$ is the GCD of all $4 A/\sqrt{r}$ values corresponding to a particular radical $r$.

Here's another picture:

dat = #[[2, 1]] & /@ radical; full = Table["", {1000}];
Do[full[[SquareFreePart[4 Hero[dat[[n]]]^2]]] = dat[[n]], {n, 1, 608}];
Grid[Prepend[Prepend[Partition[trianglepic /@ Take[full, 96], 12], Style[#, 24, Bold] & /@ {24, 8, 1, 0, 8, 8, 3, 0, 8, 24, 1, 0}], 
  Range[12]], Frame -> All]

Triangle Radicals

The attached notebook compresses data collected from millions of triangles. All Heronian triangles have area divisible by 6, based on the Brahmagupta parametric form. If this conjecture is true, that hints there might be a more powerful parametric form that classifies all integer triangles.

Does a proof exist?

For radical 953, my scalenes are {{528, 681, 697}, {533, 680, 693}, {561, 656, 689}}.
For radical 977, my scalenes are {{567, 690, 697}, {612, 657, 685}}.
Can someone find triangles with radicals 953 or 977 where $4A$ is divisible by 8, but not 16?

This came out of studying Wheel Graphs with Integer Edges. I noticed all integer-edged wheel graphs with largest edge $<10$ were composed of triangles with exactly 1 or 2 types of radical. Any counterexamples to that?

Integer wheel graph

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