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 $\eta_l>0$.

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.

Contributions to the theory of diophantine equations I. On the representation of integers by binary forms

1968 ◽  
Vol 263 (1139) ◽  
pp. 173-191 ◽  
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Binary Form ◽  
Effective Algorithm ◽  
Algebraic Numbers ◽  
Binary Forms ◽  
Integer Coefficients ◽  
Representation Of Integers

An effective algorithm is established for solving in integers x, y any Diophantine equation of the type/( x, y ) = m , where/ denotes an irreducible binary form with integer coefficients and degree at least 3. The magnitude, relative to m, of the bound furnished by the algorithm for the size of all the solutions of the equation is investigated, and, in consequence, there is obtained the first generally effective improvement on the well known result of Liouville (1844) concerning the accuracy with which algebraic numbers can be approximated by rationals.

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

2008 ◽  
Vol 60 (3) ◽  
pp. 491-519 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek
Keyword(s):  
Lower Bounds ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Arbitrary Degree ◽  
Famous Theorem ◽  
Linear Forms ◽  
Positive Integers ◽  
Level Lowering

AbstractWe solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation5uxn − 2r3s yn = ±1,in non-zero integers x, y and positive integers u, r, s and n ≥ 3. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

On the Product of Linear Forms

1943 ◽  
Vol 50 (3) ◽  
pp. 173 ◽  
Author(s):  
Gordon Pall
Keyword(s):  
Linear Forms ◽  
Product Of Linear Forms

scholarly journals HUA'S LEMMA AND SIMULTANEOUS DIAGONAL EQUATIONS

2002 ◽  
Vol 34 (3) ◽  
pp. 279-283 ◽  
Author(s):  
JÖRG BRÜDERN ◽  
TREVOR D. WOOLEY
Keyword(s):  
Asymptotic Formula ◽  
Integer Coefficients

This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [−P,P]s, provided only that s > (3r+1)2k−2. By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r2k.

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