Polynomial-Value Sieving and Recursively-Factorable Polynomials

Abstract

We identify a recursive structure among factorizations of polynomial values into two integer factors. Polynomials for which this recursive structure characterizes all non-trivial representations of integer factorizations of the polynomial values into two parts are here called recursively-factorable polynomials. In particular, we prove that $n^2+1$ and the prime-producing polynomials $n^2+n+41$ and $2n^2+ 29$ are recursively-factorable. For quadratics, the we prove that this recursive structure is equivalent to a Diophantine identity involving the product of two binary quadratic forms. We show that this identity may be transformed into geometric terms, relating each integer factorization $an^2+bn+c=pq$ to a lattice point of the conic section $aX^2+bXY+cY^2+X-nY=0$, and vice versa.