The Number in the Particle

Anomaly cancellation in quantum gauge theories imposes arithmetic constraints on particle charges. These constraints ensure the quantum theory is consistent — anomalous gauge symmetries would destroy the theory's predictive power. The constraints are sets of polynomial equations in the charges: the sum of charges to various powers must vanish.

Lee, Takahashi, and Tsai (arXiv:2603.12320) discover that the anomaly cancellation conditions for a hidden U(1) gauge sector with minicharged particles are exactly equivalent to the degree-3 Prouhet-Tarry-Escott problem — a classical question in number theory about finding two distinct sets of integers whose power sums agree up to a given degree.

The equivalence is not approximate. The physical constraint (anomaly cancellation) and the mathematical problem (equal power sums) are the same equations. Solutions to one are solutions to the other. The extensive literature on the Prouhet-Tarry-Escott problem — accumulated over more than a century of pure mathematics — becomes directly applicable to constraining what particles can exist.

The consequences are specific. The equivalence forces the hidden sector to contain at least four minicharged states — fewer than four, and no solution exists. When the minimal ideal solutions of the number-theoretic problem are used, the resulting particle spectrum has a distinctive structure: near-degenerate doublets, where particles come in pairs with identical minicharge and comparable mass. Finding one minicharged particle at a collider would predict a partner nearby in mass.

The structural lesson: the consistency conditions of physics and the structure of integers are not merely analogous — they are, in this case, identical. The same algebraic constraints that number theorists studied for their own sake turn out to govern what particles nature permits. The number theory didn't need the physics. The physics always needed the number theory. The connection was waiting to be noticed.

Lee, Takahashi, & Tsai, "Number Theory in Quantum Physics: Minicharged Particles and the Prouhet-Tarry-Escott Problem," arXiv:2603.12320 (2026).