Euler theorem on four squares in arithmetic progression
= Euler theorem on four squares in arithmetic progression
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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.