scholarly journals The Divisors of a Quadratic Polynomial

1961 ◽  
Vol 5 (1) ◽  
pp. 8-20 ◽  
Author(s):  
E. J. Scourfield
Keyword(s):  
Positive Integer ◽  
Quadratic Polynomial ◽  
Fixed Integer ◽  
Integer Coefficients

Let f(n) = an2+ bn + c be an irreducible quadratic polynomial with integer coefficients, and let D denote the discriminant b2 – 4ac of f(n).We shall assume that (D, k) = 1, and that for all positive integer n, f(n) is positive and coprime with k, where k is a fixed integer greater than 1.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

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 A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

On the decimal representation of integers

1952 ◽  
Vol 48 (4) ◽  
pp. 555-565 ◽  
Author(s):  
M. P. Drazin ◽  
J. Stanley Griffith
Keyword(s):  
Positive Integer ◽  
Fixed Integer ◽  
Usual Convention ◽  
Representation Of Integers ◽  
Image Position

Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such thatwhere, subject to the conditionsthe integers αk(r, n) are uniquely determined, and, in fact, clearlyαk(r, n) = [n/rk] − r[n/rk+1](square brackets denoting integral parts, according to the usual convention).

scholarly journals Some Natural Bigraded $S_n$-Modules

10.37236/1282 ◽  
1996 ◽  
Vol 3 (2) ◽  
Author(s):  
A. M. Garsia ◽  
M. Haiman
Keyword(s):  
Positive Integer ◽  
Linear Span ◽  
Basic Properties ◽  
Integer Coefficients ◽  
Derivatives Of

We construct for each $\mu\vdash n $ a bigraded $S_n$-module $\mathbf{H}_\mu$ and conjecture that its Frobenius characteristic $C_{\mu}(x;q,t)$ yields the Macdonald coefficients $K_{\lambda\mu}(q,t)$. To be precise, we conjecture that the expansion of $C_{\mu}(x;q,t)$ in terms of the Schur basis yields coefficients $C_{\lambda\mu}(q,t)$ which are related to the $K_{\lambda\mu}(q,t)$ by the identity $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. The validity of this would give a representation theoretical setting for the Macdonald basis $\{ P_\mu(x;q,t)\}_\mu$ and establish the Macdonald conjecture that the $K_{\lambda\mu}(q,t)$ are polynomials with positive integer coefficients. The space $\mathbf{H}_\mu$ is defined as the linear span of derivatives of a certain bihomogeneous polynomial $\Delta_\mu(x,y)$ in the variables $x_1,x_2,\ldots ,x_n$, $y_1,y_2,\ldots ,y_n$. On the validity of our conjecture $\mathbf{H}_\mu$ would necessarily have $n!$ dimension. We refer to the latter assertion as the $n!$-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials $C_{\lambda\mu}(q,t)$ have a number of basic properties in common with the coefficients $\tilde{K}_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. For instance, we show that $C_{\lambda\mu}(0,t)=\tilde{K}_{\lambda\mu}(0,t)$, $C_{\lambda\mu}(q,0)=\tilde{K}_{\lambda\mu}(q,0)$ and show that on the $n!$ conjecture we must also have the equalities $C_{\lambda\mu}(1,t)=\tilde{K}_{\lambda\mu}(1,t)$ and $C_{\lambda\mu}(q,1)=\tilde{K}_{\lambda\mu}(q,1)$. The conjectured equality $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$ will be shown here to hold true when $\lambda$ or $\mu$ is a hook. It has also been shown (see [9]) when $\mu$ is a $2$-row or $2$-column partition and in [18] when $\mu$ is an augmented hook.

Exact values of Γ∗(10,p)

2019 ◽  
Vol 16 (03) ◽  
pp. 639-649
Author(s):  
Daiane S. Veras ◽  
Paulo H. A. Rodrigues
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Diagonal Form ◽  
Integer Coefficients ◽  
Nontrivial Zero ◽  
Special Case

For [Formula: see text] and [Formula: see text] a prime number, define [Formula: see text] to be the smallest positive integer [Formula: see text] such that any diagonal form [Formula: see text], with integer coefficients, has nontrivial zero over [Formula: see text] whenever [Formula: see text]. A special case of a conjecture attributed to Artin states that [Formula: see text]. It is well known that the equality occurs when [Formula: see text]. In this paper, we obtain the exact values of [Formula: see text] for all primes [Formula: see text] and, except for [Formula: see text], these values are much lower than those established in the conjecture, as might be expected.

The Fractal Dimension of Sets Derived from Complex Bases

1986 ◽  
Vol 29 (4) ◽  
pp. 495-500 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Fractal Dimension ◽  
Positive Integer ◽  
Positive Root ◽  
Complex Numbers ◽  
Integer Part ◽  
Fixed Integer

AbstractFor each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λn/log(n2 + 1), where λn is the positive root of λ3 - (2n - 1) λ2 - (n - 1) 2λ - (n2 + 1).

scholarly journals On integers which are representable as sums of large squares

2015 ◽  
Vol 11 (08) ◽  
pp. 2505-2511 ◽  
Author(s):  
Alessio Moscariello
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Integer Coefficients ◽  
Representation Of Integers

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n2, (n + 1)2, …} is asymptotically O(n2), verifying thus a conjecture of Dutch and Rickett. Furthermore, we ask a question on the representation of integers as sum of four large squares.

scholarly journals The Number of Unitarily k-Free Divisors of An Integer

1976 ◽  
Vol 21 (1) ◽  
pp. 19-35
Author(s):  
D. Suryanarayana ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Canonical Representation ◽  
Fixed Integer ◽  
Unitary Divisor ◽  
Free Divisors

Let k be a fixed integer ≧ 2. A positive integer n is called unitarily k-free, if the multiplicity of each prime factor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the k-th power of any integer > 1. By a unitary divisor, we mean as usual, a divisor d> 0 of n such that (d, n/d) = 1. The interger 1 is also considered to be unitarily k-free. The concept of a unitarily k-free integer was first introduced by Cohen (1961; §1). Let denote the set of unitarily k-free integers. When k = 2, the set coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1960; §1 and §6). A divisor d > 0 of the positive integer n is called a unitarily k-free divisor of n if d ∈ . Let (n) denote the number of unitarily k-free divisors of n.

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