39 research outputs found

    A note on pp-adic valuations of the Schenker sums

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    A prime number pp is called a Schenker prime if there exists such nN+n\in\mathbb{N}_+ that pnp\nmid n and panp\mid a_n, where an=j=0nn!j!nja_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning pp-adic valuations of ana_n in case when pp is a Schenker prime. In particular, they asked whether for each kN+k\in\mathbb{N}_+ there exists the unique positive integer nk<pkn_k<p^k such that vp(am5k+nk)kv_p(a_{m\cdot 5^k + n_k})\geq k for each nonnegative integer mm. We prove that for every kN+k\in\mathbb{N}_+ the inequality v5(an)kv_5(a_n)\geq k has exactly one solution modulo 5k5^k. This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if 37n37\nmid n then v37(an)1v_{37}(a_n)\leq 1, 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

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    The quotient set, or ratio set, of a set of integers AA is defined as R(A):={a/b:a,bA,  b0}R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}. We consider the case in which AA is the image of Z+\mathbb{Z}^+ under a polynomial fZ[X]f \in \mathbb{Z}[X], and we give some conditions under which R(A)R(A) is dense in Qp\mathbb{Q}_p. Then, we apply these results to determine when R(Smn)R(S_m^n) is dense in Qp\mathbb{Q}_p, where SmnS_m^n is the set of numbers of the form x1n++xmnx_1^n + \cdots + x_m^n, with x1,,xm0x_1, \dots, x_m \geq 0 integers. This allows us to answer a question posed in [Garcia et al., pp-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question

    Stirling number and periodic points

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    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

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    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

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    We show that for each positive integer aa there exist only finitely many prime numbers pp such that aa appears an odd number of times in the period of continued fraction of p\sqrt{p} or 2p\sqrt{2p}. We also prove that if pp is a prime number and D=pD=p or 2p2p is such that the length of the period of continued fraction expansion of D\sqrt{D} is divisible by 44, then 11 appears as a partial quotient in the continued fraction of D\sqrt{D}. Furthermore, we give an upper bound for the period length of continued fraction expansion of D\sqrt{D}, where DD 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