The following thread is on the challenging problem of how to quickly factor a composite integer using polynomials.

I wish for other users to answer the question I make in each post, however if it goes unanswered, I'll post it myself eventually (assuming I have it) and then move on to the next part. Furthermore, I'll point out those problems which I feel are ''research questions'' that have currently gone unanswered in the literature.

Consider a degree d polynomial P(x;a) = (a^dx - 1).....(a^2x - 1)(ax - 1) and suppose n is a composite integer we wish to factor.

Let k = [\frac{\sqrt{n}}{d}] and i = a^{d} mod(n).

Argue that if gcd(a,n) = 1 then the product
P = P(i;a)P(i^2;a)P(i^3;a)....P(i^k;a)mod(n), is such that gcd(P,n) ! = 1