Factor base

A factor base is a finite set of small prime numbers over which selected auxiliary integers are completely factored. Some methods also record a sign coordinate for $-1$, which is not a prime number. A positive integer supported entirely on the chosen factor base gives an exponent vector; reducing that vector modulo two records which prime exponents are odd integers. The product is a square of a number precisely when every prime exponent is an even integer.

For the original toy $N=77$, use factor base $\{2,3,5\}$ and the two relationsThe residues have exponent vectors $(0,1,1)$ and $(2,1,1)$ in the order $(2,3,5)$. Adding the two vectors modulo two gives $(0,0,0)$, so multiplying the residues gives $15\cdot 60=900=30^2$. Consequentlyand $\gcd (X-Y,77)=7$ and $\gcd (X+Y,77)=11$. Neither individual residue was a square; combining the relations made all exponents even.

In a larger collection, choose relations $x_i^2\equiv r_i(\mathrm{mod}N)$ with positive residues $r_i={\prod }_{j=1}^B{p}_j^{e_{ji}}$. If a nonempty selection $I$ has an even sum in every exponent coordinate, setThen $X^2\equiv Y^2(\mathrm{mod}N)$. Finding such a selection is a linear dependence problem over the finite field ${\mathbb{F}}_2$: the indicator vector of the chosen relations belongs to the kernel of a linear map given by the parity matrix. This is why a problem about prime factors recruits linear algebra. The final congruence of squares may still be trivial, so another selection can be needed. ^[ref].

There is a tradeoff: a small factor base gives a smaller parity matrix, but fewer auxiliary integers factor completely over it. A larger factor base accepts more relations but makes the linear algebra larger. Negative auxiliary values need an additional sign coordinate, and a selected product needs even sign parity as well as even prime exponents. Pomerance's account explains this balance and the historical development of combining smooth relations; the toy example omits algorithm-specific sieving and running-time claims. ^[ref].