472 research outputs found
Distribution of integral values for the ratio of two linear recurrences
Let and be linear recurrences over a number field , and
let be a finitely generated subring of .
Furthermore, let be the set of positive integers such that
and . Under mild hypothesis,
Corvaja and Zannier proved that has zero asymptotic density. We
prove that
for all , where is a positive integer that can be computed in
terms of and . Assuming the Hardy-Littlewood -tuple conjecture, our
result is optimal except for the term
On the sum of digits of the factorial
Let b > 1 be an integer and denote by s_b(m) the sum of the digits of the
positive integer m when is written in base b. We prove that s_b(n!) > C_b log n
log log log n for each integer n > e, where C_b is a positive constant
depending only on b. This improves of a factor log log log n a previous lower
bound for s_b(n!) given by Luca. We prove also the same inequality but with n!
replaced by the least common multiple of 1,2,...,n.Comment: 4 page
Covering an arithmetic progression with geometric progressions and vice versa
We show that there exists a positive constant C such that the following
holds: Given an infinite arithmetic progression A of real numbers and a
sufficiently large integer n (depending on A), there needs at least Cn
geometric progressions to cover the first n terms of A. A similar result is
presented, with the role of arithmetic and geometric progressions reversed.Comment: 4 page
A note on primes in certain residue classes
Given positive integers , we prove that the set of primes
such that for admits asymptotic
density relative to the set of all primes which is at least , where is the Euler's totient
function. This result is similar to the one of Heilbronn and Rohrbach, which
says that the set of positive integer such that
for admits asymptotic density which is at least
On the greatest common divisor of and the th Fibonacci number
Let be the set of all integers of the form ,
where is a positive integer and denotes the th Fibonacci number.
We prove that for all
, and that has zero asymptotic density. Our proofs rely
on a recent result of Cubre and Rouse which gives, for each positive integer
, an explicit formula for the density of primes such that divides
the rank of appearance of , that is, the smallest positive integer such
that divides
A coprimality condition on consecutive values of polynomials
Let be quadratic or cubic polynomial. We prove that there
exists an integer such that for every integer one can
find infinitely many integers with the property that none of
is coprime to all the others. This extends
previous results on linear polynomials and, in particular, on consecutive
integers
The density of numbers having a prescribed G.C.D. with the th Fibonacci number
For each positive integer , let be the set of all positive
integers such that , where denotes the th
Fibonacci number. We prove that the asymptotic density of
exists and is equal to where is the M\"obius
function and denotes the least positive integer such that
divides . We also give an effective criterion to establish when the
asymptotic density of is zero and we show that this is the case
if and only if is empty
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