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The doubling slope at $P=(2,1)$ is $(3\cdot 2^2-1)/(2\cdot 1)=11/2$, givingThese 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 computesIt is a proper factor. The failure of the group-law division modulo the composite integer reveals different behavior modulo its prime factors.

In the elliptic curve method, addition is attempted modulo the composite number. A denominator that is not a unit in a ring supplies a factor through its greatest common divisor with that number.

First double $P$ using the elliptic-curve addition formula. Its tangent slope isThese calculations are valid modulo $221$ since the only initial denominator is two, which is invertible. Thus $2P=(14,-53)$, equivalently $(14,168)$ modulo $221$.

To form $3P=P+2P$, the chord slope would requireThe denominator hasIt cannot be inverted, and thirteen is the requested proper factor. The failure has a geometric explanation: modulo thirteen, $2P=(1,-1)=-P$, so $3P$ is the point at infinity. Modulo seventeen that chord denominator remains nonzero. The differing behavior at the two primes is exactly what the elliptic curve method detects.

Let $N=pq$ with primes $p\equiv 1(\mathrm{mod}12)$ and $q\equiv 5(\mathrm{mod}12)$. quadratic reciprocity makes three a square modulo $p$; choose $r^2=3$ and impose $a=-3+2r(\mathrm{mod}p)$, $a=1(\mathrm{mod}q)$ by the Chinese remainder theorem. The curve has good reduction at both primes. Indeed its discriminant is $-16a^2(4a+27)$; the first condition could make a factor vanish only at $p=3$ or 11, while the second gives $a(4a+27)=31$ and the given residue class excludes $q=31$.

Doubling $P$ gives $x(2P)-1=(a^2+6a-3)/4$. This vanishes modulo $p$ and equals one modulo $q$. Modulo $p$ one also has $y(2P)=-1$, so $2P=-P$ and $3P$ is the identity. Modulo $q$ the addition is finite. Thus computing $3P$ modulo $N$ encounters a denominator whose greatest common divisor with $N$ is exactly $p$, revealing the factor. The numerator difference is minus two modulo $p$, so the failed inverse cannot be cancelled. This is a construction of successful elliptic curve method data from prescribed local group behavior.