Bounded torsion of elliptic curves

An explicit bound is obtained for the torsion of elliptic curves over the field of rational numbers. Let $\Gamma$ be an elliptic curve over the field of rational numbers $R$, and $Q_m$ a primitive $R$-point of order $m$ on it; here $m$ is a prime or a double prime. Hence if $m=2p$, then $p\leqslant509$, whereas if $m=p$, then $p<6144$.