= Rational torsion of a congruent number curve
{title2=$E_D(\mathbb Q)_{\mathrm{tors}}\cong(\mathbb Z/2\mathbb Z)^2$}
For nonzero integral $D$, the <elliptic curve> $E_D:y^2=x^3-D^2x$ has rational torsion precisely $O,(0,0),(D,0),(-D,0)$. At every good <prime> $p\equiv3\pmod4$, cancellation of the <Legendre symbols> at $x$ and $-x$ gives $\#E_D(\mathbb F_p)=p+1$. The <torsion-freeness of the formal group over Qp for odd p> makes the rational <torsion subgroup> inject into this group. The <Dirichlet theorem on primes in arithmetic progressions> excludes odd factors of its order and bounds its two-primary part by four. The four displayed rational <2-torsion> points then give equality.
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