POLYNOMIAL VARIATIONS ON A THEME OF SIERPIŃSKI

2009 ◽  
Vol 05 (06) ◽  
pp. 999-1015 ◽  
Author(s):  
LENNY JONES
Keyword(s):  
Integer Coefficients ◽  
Variations On A Theme ◽  
Positive Integers

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals About the number of τ-numbers relative to polynomials with integer coefficients

2021 ◽  
Vol 25 (1) ◽  
pp. 107-117
Author(s):  
Mart Abel ◽  
Helena Lauer ◽  
Ellen Redi
Keyword(s):  
Extra Condition ◽  
Integer Coefficients ◽  
Positive Integers

We show that for all polynomials Q(x) with integer coefficients, that satisfy the extra condition |Q(0) · Q(1) | ≠ 1, there are infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). We also find some examples of polynomials Q(x) for which 1 is the only τ-number relative to the polynomial Q(x) and some examples of polynomials Q(x) with |Q(0) · Q(1)|= 1, which have infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). In addition, we prove one result about the generators of a τ-number.

scholarly journals Three additive congruences to a large prime modulus

1993 ◽  
Vol 55 (3) ◽  
pp. 355-368
Author(s):  
O. D. Atkinson ◽  
J. Brüdern ◽  
R. J. Cook
Keyword(s):  
Positive Constant ◽  
Integer Coefficients ◽  
Prime Modulus ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 3 and n > 6k be positive integers. The equations, with integer coefficients, have nontrivial p-adic solutions for all p > Ck8, where C is some positive constant. Further, for values k≥ K we can take C = 1 + O(K-½).

Intervals without primes near elements of linear recurrence sequences

2018 ◽  
Vol 14 (02) ◽  
pp. 567-579
Author(s):  
Artūras Dubickas
Keyword(s):  
Recurrence Relation ◽  
Arithmetic Progression ◽  
Prime Numbers ◽  
Linear Recurrence ◽  
Integer Coefficients ◽  
Salem Numbers ◽  
Consecutive Integers ◽  
Linear Recurrence Relation ◽  
Recurrence Sequences ◽  
Positive Integers

Let [Formula: see text] be an unbounded sequence of integers satisfying a linear recurrence relation with integer coefficients. We show that for any [Formula: see text] there exist infinitely many [Formula: see text] for which [Formula: see text] consecutive integers [Formula: see text] are all divisible by certain primes. Moreover, if the sequence of integers [Formula: see text] satisfying a linear recurrence relation is unbounded and non-degenerate then for some constant [Formula: see text] the intervals [Formula: see text] do not contain prime numbers for infinitely many [Formula: see text]. Applying this argument to sequences of integer parts of powers of Pisot and Salem numbers [Formula: see text] we derive a similar result for those sequences as well which implies, for instance, that the shifted integer parts [Formula: see text], where [Formula: see text] and [Formula: see text] runs through some infinite arithmetic progression of positive integers, are all composite.

scholarly journals On totally reducible binary forms: II.

2002 ◽  
Vol Volume 25 ◽  
Author(s):  
C Hooley
Keyword(s):  
Asymptotic Formula ◽  
Principal Result ◽  
Binary Form ◽  
Binary Forms ◽  
Linear Forms ◽  
Integer Coefficients ◽  
International Audience ◽  
Product Of Linear Forms ◽  
Positive Integers

International audience Let $f$ be a binary form of degree $l\geq3$, that is, a product of linear forms with integer coefficients. The principal result of this paper is an asymptotic formula of the shape $n^{2/l}(C(f)+O(n^{-\eta_l+\varepsilon}))$ for the number of positive integers not exceeding $n$ that are representable by $f$; here $C(f)>0$ and $\eta_l>0$.

Universal diophantine equation

1982 ◽  
Vol 47 (3) ◽  
pp. 549-571 ◽  
Author(s):  
James P. Jones
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
Positive Integers ◽  
Martin Davis ◽  
Exponential Relation

In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the formHere P is a polynomial with integer coefficients and the variables range over positive integers.In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved [11] that the exponential relation y = 2x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the formFrom this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets.Now it is well known that the recursively enumerable sets W1, W2, W3, … can be enumerated in such a way that the binary relation x ∈ Wv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,

Clara Schumann's Interiorities and the Cutting Edge of Popular Pianism

2017 ◽  
Vol 70 (3) ◽  
pp. 697-765
Author(s):  
Alexander Stefaniak
Keyword(s):  
Cutting Edge ◽  
Critical Discourse ◽  
Robert Schumann ◽  
Piano Concerto ◽  
Sheet Music ◽  
Variation Sets ◽  
Music Market ◽  
Critical Responses ◽  
Drawing Room ◽  
Variations On A Theme

In her contemporaries’ imaginations Clara Schumann transcended aesthetic pitfalls endemic to virtuosity. Scholars have stressed her performance of canonic repertory as a practice through which she established this image. In this study I argue that her concerts of the 1830s and 1840s also staged an elevated form of virtuosity through showpieces that inhabited the flagship genres of popular pianism and that, for contemporary critics, possessed qualities of interiority that allowed them to transcend merely physical or “mechanical” engagement with virtuosity. They include Henselt's études and variation sets, Chopin's “Là ci darem” Variations, op. 2, and Clara's own Romance variée, op. 3, Piano Concerto, op. 7, and Pirate Variations, op. 8. Her 1830s and early 1840s programming offers a window onto a rich intertwining of critical discourse, her own and her peers’ compositions, and her strategies as a pianist-composer. This context reveals that aspirations about elevating virtuosity shaped a broader, more varied field of repertory, compositional strategies, and critical responses than we have recognized. It was a capacious, flexible ideology and category whose discourses pervaded the sheet music market, the stage, and the drawing room and embraced not only a venerated, canonic tradition but also the latest popularly styled virtuosic vehicles. In the final stages of the article I propose that Clara Schumann's 1853 Variations on a Theme by Robert Schumann, op. 20, alludes to her work of the 1830s and 1840s, evoking the range of guises this pianist-composer gave to her virtuosity in what was already a wide-ranging career.

Brahms, Rhythm, and the Renaissance

2012 ◽  
Vol 8 ◽  
Author(s):  
Eleanor Heisey
Keyword(s):  
Compositional Style ◽  
The Past ◽  
Piano Quartet ◽  
Look Back ◽  
Variations On A Theme

Johannes Brahms’s deep engagement with the past contributed to his compositional style in many ways. This article considers Brahms techniques that look back to and expand on those of Renaissance composers, in particular metric conflict and cadences, voice displacement, changes in proportion, rhythmic augmentation and diminution, and the hocket. Examples are taken from Brahms’s Academic Festival Overture, Variations On A Theme By Haydn, Piano Quartet in A Major, and Symphony No. 3 in F Major.

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