Polynomial-Value Sieving and Recursively-Factorable Polynomials

الأصل · EN

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 n²+1 and the prime-producing polynomials n²+n+41 and 2n²+ 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 an²+bn+c=pq to a lattice point of the conic section aX²+bXY+cY²+X-nY=0, and vice versa.