Number Theory in Quantum Physics: Minicharged Particles and the Prouhet-Tarry-Escott Problem

In quantum gauge theories, anomaly cancellation severely restricts the allowed patterns of chiral charges. Here we show that, in a phenomenologically motivated framework for light minicharged particles, the anomaly cancellation conditions are equivalent to the degree $k=3$ Prouhet-Tarry-Escott problem in number theory. This correspondence immediately implies that the hidden sector must contain at least four minicharged states. For constructions based on minimal ideal solutions, the mass spectrum generically exhibits a near-degenerate doublet structure, so that the discovery of one minicharged particle would point to a partner state with the same minicharge and a nearby mass. Our results uncover an unexpected link between quantum consistency and number theory, with direct implications for model building and future searches.