Mordell-Weil group of y squared equals x times x plus one times x plus four ($E(\mathbb{Q})$ for $E:y^2=x(x+1)(x+4)$)

For $E:y^2=x(x+1)(x+4)$,The points $(-2,2)$ and $(-1,0)$ generate the two factors, and a two-descent on an elliptic curve proves that the Mordell-Weil group has rank zero.

There is no nonconstant four-term arithmetic progression of rational square numbers. If $r^2,s^2,1,t^2$ formed one, then $(-2s^2,2rst)$ would lie on $y^2=x(x+1)(x+4)$; its explicitly known Mordell-Weil group forces $s^2=1$ and the common difference to vanish.