On integers nn for which Xn−1X^n-1 has a divisor of every degree

A positive integer $n$ is called $\varphi$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $\varphi$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$

Primitive sets with large counting functions

A set of positive integers is said to be primitive if no element of the set is a multiple of another. If $S$ is a primitive set and $S(x)$ is the number of elements of $S$ not exceeding $x$, then a result of Erd\H os implies that $\int_2^\infty (S(t)/t^2\log t) dt$ converges. We establish an approximate converse to this theorem, showing that if $F$ satisfies some mild conditions and $\int_2^\infty (F(t)/t^2\log t) dt$ converges, then there exists a primitive set $S$ with $S(x) \gg F(x)$.Comment: 7 pages. Revision includes a strengthening of Theorem 1: an upper bound for S(x) of the same order of magnitude as the lower bound is now establishe

The iterated Carmichael \lambda-function and the number of cycles of the power generator

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to \lambda(\lambda(n)), where \lambda(n) denotes Carmichael's function, namely the maximal multiplicative order of any integer modulo n. In this paper, we show that for almost all n, the size of \lambda(\lambda(n)) is n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula for the k-th iterate of \lambda. We deduce that for almost all n, the psuedorandom number generator described above has at least exp((1+o(1))(log log n)^2 log log log n) disjoint cycles. In addition, we show that this expression is accurate for almost all n under the assumption of the Generalized Riemann Hypothesis for Kummerian fields. We also consider the number of iterations of \lambda it takes to reduce an integer n to 1, proving that this number is less than (1+o(1))(log log n)/log 2 infinitely often and speculating that log log n is the true order of magnitude almost always.Comment: 28 page

Squarefree smooth numbers and Euclidean prime generators

We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.Comment: 8 pages, to appear in Proceedings of the AM

Long gaps in sieved sets

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^{\delta}$ where $\delta>0$ depends only on the density of primes for which $I_p\ne \emptyset$. This improves on the ``trivial'' bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{ n \leq X: f(n) \hbox{ composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^{\delta}$ for sufficiently large $X$, where $\delta>0$ depends only on the degree of $f$.Comment: Major revision. We replaced the PNT-type assumption with (a) a Mertens estimate; (b) that the density $\rho$ of nonempty $I_p$ exists. Our main theorem now gives an exponent which is a function of $\rho$, and is completely explicit. In particular, the exponent $e^{-1-4/\rho}$ is admissible. Various notational simplifications. Many remarks added to help the reade