scholarly journals Arithmetic properties of Apéry-like numbers

2017 ◽  
Vol 154 (2) ◽  
pp. 249-274 ◽  
Author(s):  
É. Delaygue
Keyword(s):  
Recurrence Relation ◽  
Lower Bounds ◽  
Finite Alphabet ◽  
Laurent Polynomials ◽  
Arithmetic Properties ◽  
Apéry Numbers ◽  
Adic Valuation ◽  
Almost All

We provide lower bounds for$p$-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the$p$-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the$p$-Lucas property for almost all primes$p$.

scholarly journals The Cube Recurrence

10.37236/1826 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Gabriel D. Carroll ◽  
David Speyer
Keyword(s):  
Recurrence Relation ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Laurent Polynomials ◽  
Combinatorial Interpretation ◽  
Combinatorial Model

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ${\Bbb Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

The geometric unfolding of recurrence relations

2020 ◽  
Vol 104 (561) ◽  
pp. 403-411
Author(s):  
Stan Dolan
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Initial Values ◽  
Almost All

In 1942 R. C. Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values [1].

scholarly journals Further Results on a Curious Arithmetic Function

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Long Chen ◽  
Kaimin Cheng ◽  
Tingting Wang
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Rational Numbers ◽  
Arithmetic Properties ◽  
Adic Valuation ◽  
Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

scholarly journals Primes in Short Segments of Arithmetic Progressions

1998 ◽  
Vol 50 (3) ◽  
pp. 563-580 ◽  
Author(s):  
D. A. Goldston ◽  
C. Y. Yildirim
Keyword(s):  
Asymptotic Formula ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Riemann Hypothesis ◽  
Total Variance ◽  
Correct Order ◽  
Asymptotic Results ◽  
Extended Range ◽  
Order Of Magnitude ◽  
Almost All

AbstractConsider the variance for the number of primes that are both in the interval [y,y + h] for y ∈ [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 ≤ h/q ≤ x1/2-∈ , for any ∈ > 0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all” q in the range 1 ≤ h/q ≤ x1/4-∈, that on averaging over q one obtains an asymptotic formula in the extended range 1 ≤ h/q ≤ x1/2-∈, and that there are lower bounds with the correct order of magnitude for all q in the range 1 ≤ h/q ≤ x1/3-∈.

scholarly journals A lower bound for general t-stack sortable permutations via pattern avoidance

2020 ◽  
Vol 22 ◽  
Author(s):  
Pranav Chinmay
Keyword(s):  
Lower Bound ◽  
Recurrence Relation ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Pattern Avoidance ◽  
Lower And Upper Bounds

There is no formula for general t-stack sortable permutations. Thus, we attempt to study them by establishing lower and upper bounds. Permutations that avoid certain pattern sets provide natural lower bounds. This paper presents a recurrence relation that counts the number of permutations that avoid the set (23451,24351,32451,34251,42351,43251). This establishes a lower bound on 3-stack sortable permutations. Additionally, the proof generalizes to provide lower bounds for all t-stack sortable permutations.

scholarly journals Exponentially Small Lower Bounds for the Splitting of Separatrices to Whiskered Tori with Frequencies of Constant Type

2014 ◽  
Vol 24 (08) ◽  
pp. 1440011 ◽  
Author(s):  
Amadeu Delshams ◽  
Marina Gonchenko ◽  
Pere Gutiérrez
Keyword(s):  
Lower Bounds ◽  
Invariant Manifolds ◽  
Irrational Number ◽  
Melnikov Method ◽  
Arithmetic Properties ◽  
Constant Type ◽  
Whiskered Tori ◽  
Vector Formula ◽  
Splitting Of Invariant Manifolds ◽  
Exponentially Small

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a two-dimensional torus with a fast frequency vector [Formula: see text], with ω = (1, Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.

scholarly journals Fairtrade and Market Efficiency: Fairtrade-Labeled Coffee in the Swedish Coffee Market

Economies ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 30
Author(s):  
Dick Durevall
Keyword(s):  
Market Efficiency ◽  
Lower Bounds ◽  
Production Costs ◽  
Upper And Lower Bounds ◽  
Scanner Data ◽  
Coffee Farmers ◽  
Additional Costs ◽  
Almost All ◽  
The Cost ◽  
Ground Coffee

Fairtrade labeling has the potential to increase market efficiency by connecting farmers to altruistic consumers who are willing to pay a premium for sustainability-certified products. A requirement for increased efficiency, though, is that the farmers’ benefits are larger than the Fairtrade processing costs and the excess payment by consumers that does not accrue to farmers; otherwise direct transfers to farmers would be more efficient. This paper analyzes how excess payment for Fairtrade-labeled coffee is distributed in the Swedish market, using information on production costs and scanner data on almost all roasted and ground coffee products sold by retailers. A key finding is that roasters and retailers get 61–70%, while producer countries, in this paper comprising coffee farmers, cooperatives, middlemen, exporters, and Fairtrade International, get 24–31%; Fairtrade Sweden gets 6–8%. These values are the upper and lower bounds that reflect assumptions made about the additional costs of producing roasted and ground Fairtrade coffee, given the cost of beans and the Fairtrade license. The Fairtrade label thus seems to create a coffee product that roasters and retailers can use to exploit their market power.

On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

2016 ◽  
Vol 164 (1) ◽  
pp. 67-98 ◽  
Author(s):  
YUKAKO KEZUKA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Complex Multiplication ◽  
Ring Of Integers ◽  
Quadratic Twists ◽  
The Family ◽  
Adic Valuation

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

scholarly journals Arithmetic Properties of Overpartition Pairs into Odd Parts

10.37236/2274 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Lishuang Lin
Keyword(s):  
Positive Integer ◽  
Arithmetic Properties ◽  
Powers Of 2 ◽  
Fixed Positive Integer ◽  
Almost All ◽  
Small Powers

In this work, we investigate various arithmetic properties of the function $\overline{pp}_o(n)$, the number of overpartition pairs of $n$ into odd parts. We obtain a number of Ramanujan type congruences modulo small powers of $2$ for $\overline{pp}_o(n)$. For a fixed positive integer $k$, we further show that $\overline{pp}_o(n)$ is divisible by $2^k$ for almost all $n$. We also find several infinite families of congruences for $\overline{pp}_o(n)$ modulo $3$ and two formulae for $\overline{pp}_o(6n+3)$  and  $\overline{pp}_o(12n)$ modulo $3$.

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