Relations from i to geometry

Saw this video from Numberphile a few weeks ago:

The boys and I had some fun talking through a few of the examples. For instance, we saw that 5 is not a prime in the Gaussian integers because 5 = (2 + i)(2 – i).

We also saw that the set of integers combined with integer multiples of $\sqrt{2}$ has some unexpected factorization properties, for example, like 2 = (2 – $\sqrt{2}$)*(2 + $\sqrt{2}$), and 14 = 2 * 7 = (4 – $\sqrt{2}$)(4 + $\sqrt{2}$). We also played with the $\sqrt{-5}$ example in the video.

I spent the last few days kicking around a few ideas about how to explore these ideas in other ways with the boys. Part of the struggle was figuring out how to translate advanced math ideas like the polynomial ring Z($x$] / ($x^2 + 1$) into something that the boys could understand.

I couldn’t crack the code for that, but thinking about this idea led to a fun coincidence when I saw this problem from Matt Enlow today:

I won’t give away the precise solution to the problem, but my approach used the idea of a polynomial relation. A simple relation like $x^5 = 2x^3 + x$ (but not that exact relation) arose from the conditions in the problem and that relation allowed me to solve Enlow’s problem without actually having to solve for $x$.

It is neat to see similar algebraic ideas arising in totally different contests.