It turns out a 200-year-old math puzzle is actually the secret rulebook for how many different types of particles can exist in our universe.
arXiv · March 16, 2026 · 2603.12320
Why it matters
The Prouhet-Tarry-Escott problem is a puzzle in number theory about sets of integers with equal powers. This paper proves that the quantum consistency of new 'minicharged' particles is governed by this exact math, meaning if we find one such particle, we can use 18th-century number theory to predict its partners and their masses.
From the abstract
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