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

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

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

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 $x \rightarrow \infty$

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

AbstractSzemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ⩾ n(k, B) and 0 < a1 < … < an is a sequence of integers with an ⩽ Bn, then some k of the ai form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ⩾ m(k, B) and u0, u1, …, um is a sequence of plane lattice points with ∑i=1m…ui − ui−1… ⩽ Bm, then some k of the ui are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result