This talk on Apollonian circle packings is great: https://youtu.be/3mU9d0cPgGM
The local-global conjecture said that other than the "obvious" congruence restrictions, every sufficiently large natural number appeared as the curvature in a circle packing. But this is false and the counterexample is not at all hard: they exhibit a circle packing that contains no squares, and a simple proof based on reciprocity.
The best part is that this started with a REU (an undergrad) doing some visualizations and finding an increasingly-clear pattern suggesting the conjecture couldn't be true.
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