Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 3 / v / Solution

Use the stated rational change of variablesDirect substitution turns the anomaly equation intoWriting $v=a/c$ and $w=b/c$ in lowest common denominator gives the integer equation $a^3+b^3=c^3$. The supplied special case of Fermat's Last Theorem says that one of $a,b$ must vanish. Thus $(v,w)=(1,0)$ or $(0,1)$, which givesThe two signs merely exchange the names of the up- and down-type singlets. Choosing $q=1$ and the conventional sign $y=3$ gives the unique assignment up to overall scaling and that exchange:These are six times the conventional Standard Model hypercharges.