---

Statement

For every $t\in\mathbb Q$, $(t^2-2)(t^2-3)$ is not a square in $\mathbb Q$.

Source and scope. Beeson–Laczkovich–Zhang, arXiv:2604.03609v3, Proposition 20, pp. 10–11. Complete rewritten reduction using the explicit external rank/torsion input below.

Proof

Suppose $s^2=t^4-5t^2+6$. By Lemma 18, $x=2t^2-2s-5$, $y=2tx$ is an affine rational point of

$$E:\quad y^2=x^3+10x^2+x.$$

The source imports rank zero and torsion order two, also recorded by LMFDB 96.b1. Its model $y^2=X^3+X^2-32X+60$ uses $X=x+3$. Consequently the only rational points are $\mathcal O$ and $(0,0)$. The affine image must be $(0,0)$, so $s=t^2-5/2$. Squaring gives $s^2=t^4-5t^2+25/4$, inconsistent with the assumed constant term $6$. This contradiction proves the claim.

Dependency boundary. The rank/torsion calculation is an external database input, as in Proposition 19, not a descent proof supplied here.

Bears on. Problem 633.