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arXiv · 2508.05347

Squares, three fleas, sporadic sets, and squares

Giedrius Alkauskas
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Abstract

Three fleas are on the plane. At each move, two fleas jump to the other two vertices of a square erected on the segment joining them. Starting from $\{(0,0),(2,1),(3,2)\}$, we ask which areas can occur for the triangle they span. Every positive half-integer occurs, and the attainable integer areas meet every residue class modulo every modulus. Yet no perfect square ever occurs. We explain this unexpected obstruction using quadratic reciprocity, and then uncover its geometric source: the flea dynamics are closely related to integral Apollonian circle packings. An exact computation up to $10^{12}$ finds only five additional missing nonsquares: $5,29,80,99,179$.

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Giedrius Alkauskas. 2026-09-09. Squares, three fleas, sporadic sets, and squares. https://arxiv.org/abs/2508.05347

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