= Solution
The doubling slope at $P=(2,1)$ is $(3\cdot2^2-1)/(2\cdot1)=11/2$, giving
$$
2P=\left(\frac{105}{4},-\frac{1075}{8}\right).
$$
These denominators are invertible modulo $N$. To add $P$ again, the difference of first coordinates is $105/4-2=97/4$, so the new slope would be $-1083/194$. This requires inversion of a number divisible by 97. The <elliptic curve method> instead computes
$$
\boxed{\gcd(97,10573)=97,\qquad10573=97\cdot109.}
$$
It is a proper factor. The failure of the group-law division modulo the composite <integer> reveals different behavior modulo its prime factors.
Back to article page
