Prime factors

Prime type III factors.

It is shown that for each 0 < lambda < 1, the free Araki-Woods factor of type III(lambda) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime) and does not contain a Cartan subalgebra.

Prime factors of consecutive integers

This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds k. This algorithm enables us to show that f(k) is not monotone, verifying a conjecture of Ecklund and Eggleton.

Prime Factors of Dynamical Sequences

Let φ(t) ∈ Q(t) have degree d ≥ 2. For a given rational number x0, define xn+1 = φ(xn) for each n ≥ 0. If this sequence is not eventually periodic, and if φ does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then xn+1 − xn has a primitive prime factor in its numerator for all sufficiently large n. The same result holds for the exceptional maps provi...

Polynomial values free of large prime factors

For F ∈ Z[X], let ΨF (x, y) denote the number of positive integers n not exceeding x such that F (n) is free of prime factors > y. Our main purpose is to obtain lower bounds of the form ΨF (x, y) x for arbitrary F and for y equal to a suitable power of x. Our proofs rest on some results and methods of two articles by the third author concerning localization of divisors of polynomial values. Ana...

Some Remarks on Prime Factors of Integers