Most factorization algorithms in use today fit in one of two camps: sieve-based methods based on congruences of squares, and algorithms based on decompositions of algebraic groups. In this article, we trace the common thread connecting the latter.
One of Landau’s four problems from 1912 concerns the infinitude of primes in the values of a certain quadratic polynomial.
In this post, we show that the largest prime factors of the values of this polynomial are “relatively large” infinitely often.
Dirichlet’s Theorem on the infinitude of primes in arithmetic progressions relies on the non-vanishing of non-trivial Dirichlet characters at 1.
In this post, I’ll show how this reduction can be introduced in an intuitive way via sieve theory. If we actually sieve, we obtain estimates for the number of integers whose prime factors lie in given congruence classes.