Tootfinder

On certain root number $1$ cases of the cube sum problem
Shamik Das, Somnath Jha
https://arxiv.org/abs/2508.05361 https://arxiv.org/pdf/2508.05361

On certain root number $1$ cases of the cube sum problem
We consider certain families of integers $n$ determined by some congruence condition, such that the global root number of the elliptic curve $E_{-432n^2}: Y^2=X^3-432n^2$ is $1$ for every $n$, however a given $n$ may or may not be a sum of two rational cubes. We give explicit criteria in terms of the $2$-parts and $3$-parts of the ideal class groups of certain cubic number fields to determine whether such an $n$ is a cube sum. In particular, we study integers $n$ divisible by $3$ such that the …

Fermion mass ratios from the exceptional Jordan algebra
Tejinder P. Singh
https://arxiv.org/abs/2508.10131 https://arxiv.org/pdf/2508.10131

Fermion mass ratios from the exceptional Jordan algebra
We revisit the "three generations" problem and the pattern of charged-fermion masses from the vantage of octonionic and Clifford algebra structures. Working with the exceptional Jordan algebra J3(OC) (right-handed flavor) and the symmetric cube of SU(3) (left-handed charge frame), we show that a single minimal ladder in the symmetric cube, together with the Dynkin Z2 swap (the A2 diagram flip), leads to closed-form expressions for the square-root mass ratios of all three charged families. The u…