= Two-prime bound on rational elliptic torsion
Let an <elliptic curve> over $\mathbb Q$ have <good reduction> at distinct primes $p,r$, with point counts $N_p,N_r$. For a torsion point, its $\ell$-primary part injects into every reduction whose residue characteristic differs from $\ell$. Consequently its $\ell$-power order has exponent at most $\min(v_\ell(N_p),v_\ell(N_r))$ when $\ell\ne p,r$, at most $v_p(N_r)$ when $\ell=p$, and at most $v_r(N_p)$ when $\ell=r$. This precise form avoids assuming injectivity on residue-characteristic torsion. Counts five at two and seven at three force trivial rational torsion.
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