185 research outputs found
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 is a primitive set and is the number of
elements of not exceeding , then a result of Erd\H os implies that
converges. We establish an approximate
converse to this theorem, showing that if satisfies some mild conditions
and converges, then there exists a
primitive set with .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 for which has a divisor of every degree
A positive integer is called -practical if the polynomial
has a divisor in of every degree up to . In this
paper, we show that the count of -practical numbers in is
asymptotic to for some positive constant as
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
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