# Hypercharge Quantisation and Fermat's Last Theorem

## TL;DR

This paper links hypercharge quantisation in the Standard Model to solutions of Fermat's Last Theorem, showing that anomaly cancellation conditions imply hypercharges are quantised and match observed values.

## Contribution

It demonstrates that anomaly cancellation constraints can be expressed as a Fermat-like equation, connecting hypercharge quantisation to a famous number theory problem.

## Key findings

- Hypercharge values correspond to integer solutions of x^3 + y^3 = z^3.
- The only solutions reproduce observed hypercharge assignments.
- Hypercharge quantisation ensures anomaly cancellation without gravitational anomaly considerations.

## Abstract

What values of the Standard Model hypercharges result in a mathematically consistent quantum field theory? We show that the constraints imposed by the lack of gauge anomalies can be recast as the equation x^3 + y^3 = z^3. If hypercharge is quantised, then x, y and z must be integers. The trivial (and only) solutions, with x=0 or y=0, reproduce the hypercharge assignments seen in Nature. This argument does not rely on the mixed gauge-gravitational anomaly, which is automatically vanishing if hypercharge is quantised and the gauge anomalies vanish.