| We address conjectures of Paul Erdős and conjectures of Yong-Gao Chen concerning Sierpinski numbers, Riesel numbers and a conjecture of de Polignac. We obtain a variety of related results, including a new smallest known positive integer that is simultaneously a Sierpinski number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k, k 2, k3,..., kr are simultaneously Sierpinski numbers.; We use covering arguments to construct an infinite family of composite, natural numbers, coprime to 10, each with the property that if you replace any one of its digits with x ∈ {lcub}0,...,9{rcub}, then the new number created by this replacement is composite. We also construct an infinite family of composite numbers, coprime to 10, each with the property that if you insert any digit x ∈ {lcub}0,...,9{rcub} into its decimal expansion, then the new number created by this insertion is composite.; Let f(x) be a monic polynomial in Z [x] of degree d > 1. We give a proof that the number reals(y) of representations of f(x) as a sum of two irreducible monic polynomials g(x) and h(x) in Z [x], with the coefficients of g( x) and h(x) bounded in absolute value by y, is asymptotic to (2y) d-1. |
