47 research outputs found
A note on -adic valuations of the Schenker sums
A prime number is called a Schenker prime if there exists such
that and , where is so-called Schenker sum. T. Amdeberhan, D.
Callan and V. Moll formulated two conjectures concerning -adic valuations of
in case when is a Schenker prime. In particular, they asked whether
for each there exists the unique positive integer
such that for each nonnegative integer .
We prove that for every the inequality has
exactly one solution modulo . This confirms the first conjecture stated by
the mentioned authors. Moreover, we show that if then
, what means that the second conjecture stated by the
mentioned authors is not true
On the p-adic denseness of the quotient set of a polynomial image
The quotient set, or ratio set, of a set of integers is defined as . We consider the case in which
is the image of under a polynomial ,
and we give some conditions under which is dense in .
Then, we apply these results to determine when is dense in
, where is the set of numbers of the form , with integers. This allows us to answer a
question posed in [Garcia et al., -adic quotient sets, Acta Arith. 179,
163-184]. We end leaving an open question
Stirling number and periodic points
We introduce the notion of almost realizability, an arithmetic generalization
of realizability for integer sequences, which is the property of counting
periodic points for some map. We characterize the intersection between the set
of Stirling sequences (of both the first and the second kind) and the set of
almost realizable sequences.Comment: 13 page
On some properties of the number of permutations being products of pairwise disjoint d-cycles
Let d≥2 be an integer. In this paper we study arithmetic properties of the sequence (Hd(n))n∈N, where Hd(n) is the number of permutations in Sn being products of pairwise disjoint cycles of a fixed length d. In particular we deal with periodicity modulo a given positive integer, behaviour of the p-adic valuations and various divisibility properties. Moreover, we introduce some related families of polynomials and study their properties. Among many results we obtain qualitative description of the p-adic valuation of the number Hd(n) extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida
On continued fraction partial quotients of square roots of primes
We show that for each positive integer there exist only finitely many
prime numbers such that appears an odd number of times in the period of
continued fraction of or . We also prove that if is a
prime number and or is such that the length of the period of
continued fraction expansion of is divisible by , then
appears as a partial quotient in the continued fraction of .
Furthermore, we give an upper bound for the period length of continued fraction
expansion of , where is a positive non-square, and factorize some
family of polynomials with integral coefficients connected with continued
fractions of square roots of positive integers. These results answer several
questions recently posed by Miska and Ulas.Comment: 14 pages, to appear in JN