Minimally Intersective Polynomials with Arbitrarily Many Quadratic Factors

Abstract

Given a natural number $n\ge 4$ we show that there exists infinitely many polynomials $f_n(x):={\prod }_{i=1}^n(x^2-a_i)$ such that (i) $f_n(x)$ has a root modulo every positive integer, (ii) $f_n(x)$ has no rational roots, and (iii) every proper divisor of $f_n(x)$ fails to have root modulo some positive integer. We exhibit a process to explicitly construct such $f_n$ and this process demonstrates that the set of natural numbers $a_n$, such that the polynomial $f_n(x):={\prod }_{i=1}^n(x^2-a_i)$ satisfies the properties (i), (ii) and (iii), is of positive asymptotic density in $\mathbb{N}$.

arXiv commentImproved exposition and correction of some computational errors

References in corpus (1)

• Polynomials Consisting of Quadratic Factors with Roots Modulo Any Positive Integer