On Friday, July 6, Melanie Palma and I began exploring ways to use a circle to factor integers into two Gaussian integers. For example, to factor 10 into Gaussian integers, we could use a circle centered at (5,0) and passing through the origin and (10,0). This circle contains ten other Gaussian integers, and we could use these to help us factor 10.
Our initial conjecture was these Gaussian integers on the circle would be factors of 10. We found that only a couple of them were. Melanie and I began to look for other factors inside the circle. We found these factors lie on chords of the circle connecting the origin to the Gaussian integers on the circle.
We noticed a great deal of symmetry in the factors, finding that pairs of factors of 10, along with their conjugates, formed the vertices of isosceles trapezoids. Being isosceles trapezoids are cyclic, our next conjecture were these quartets of points would lie on concentric circles centered at (5,0).
This conjecture turned out not to be false. In fact, we found a couple of factors appeared not to be cyclic. We formed a new conjecture at the end of our problem-solving question: The factor pairs of 10, and their conjugates, lie on Apollonian Circles.
We left it at that. As I explored the problem further that afternoon, I became convinced that these circles were indeed Apollonian, and I conjectured they would accumulate around the roots of 10. I developed a proof of this over the following weekend, and shared the preliminary results with the Working Group on Monday, June 8.
I have since decided to make this the focus of my studies in the Working Group, and shared my results to date with everyone in the High School Teachers Program this morning.
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