On generalised Fibonaccian and Lucasian numbers

2006 ◽  
Vol 90 (518) ◽  
pp. 215-222 ◽  
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Residue Class ◽  
The Other ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Residue Classes ◽  
Other Hand ◽  
Positive Integers ◽  
Striking Fact

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number L n. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

scholarly journals n-Homogeneous C*-algebras

2021 ◽  
Vol 57 (2) ◽  
pp. 185-189
Author(s):  
M. V. Shchukin
Keyword(s):  
Residue Class ◽  
The Other ◽  
Two Dimensional ◽  
3 Dimensional ◽  
C Algebras ◽  
Residue Classes ◽  
Other Hand ◽  
Primitive Ideals

The classical results by J. Fell, J. Tomiyama, M. Takesaki describe n-homogeneous С*-algebras as algebras of all continuous sections for an appropriate algebraic bundle. By using this realization, several authors described the set of n-homogeneous С*-algebras with different spaces of primitive ideals. In 1974 F. Krauss and T. Lawson described the set of all n-homogeneous С*-algebras whose space Prim of primitive ideals is homeomorphic to the sphere S2. Suppose the space PrimA of primitive ideals is homeomorphic to the sphere S3 for some n-homogeneous С*-algebra A. In this case, these authors showed that the algebra A is isomorphic to the algebra C(S3,Cn×n). If n ≥ 2 then there are countably many pairwise non-isomorphic n-homogeneous С*-algebras A such that PrimA ≅ S 4. Further, let n ≥ 3. There is only one n-homogeneous С*-algebra A such that PrimA ≅ S 5. There are two non-isomorphic 2-homogeneous С*-algebras A and B with space PrimA ≅ S 5. On the other hand, algebraic bundles over the torus T 2 are described by a residue class [p] in Z/nZ = π1(PUn). Two such bundles with classes [pi] produce isomorphic С*-algebras if and only if [p1] = ±[p2]. An algebraic bundle over the torus T 3 is determined by three residue classes in Z/nZ. Anatolii Antonevich and Nahum Krupnik introduced some structures on the set of algebraic bundles over the sphere S2. Algebraic bundles over the compact connected two-dimensional oriented manifolds were considered by the author. In this case, the set of non-equivalent algebraic bundles over such space is like the set of algebraic bundles over the torus T2. Further advances could be in describing the set of algebraic bundles over the 3-dimensional manifolds.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

Arithmetic functions monotonic at consecutive arguments

2014 ◽  
Vol 51 (2) ◽  
pp. 155-164
Author(s):  
Jean-Marie Koninck ◽  
Florian Luca
Keyword(s):  
Large Class ◽  
The Other ◽  
Arithmetic Functions ◽  
Arbitrary Integer ◽  
Other Hand ◽  
Positive Integers

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

scholarly journals Finite groups of deficiency zero involving the Lucas numbers

1990 ◽  
Vol 33 (1) ◽  
pp. 1-10 ◽  
Author(s):  
C. M. Campbell ◽  
E. F. Robertson ◽  
R. M. Thomas
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Fibonacci Number ◽  
Single Parameter ◽  
The Other ◽  
Finite Soluble Group ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Derived Length ◽  
New Class

In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

scholarly journals On the size of Diophantine m-tuples

2002 ◽  
Vol 132 (1) ◽  
pp. 23-33 ◽  
Author(s):  
ANDREJ DUJELLA
Keyword(s):  
The Other ◽  
Famous Conjecture ◽  
Nonzero Integer ◽  
Other Hand ◽  
Parametric Families ◽  
Diophantine Quintuples ◽  
Positive Integers

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

scholarly journals Discrepancy of Matrices of Zeros and Ones

10.37236/1447 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
R. A. Brualdi ◽  
J. Shen
Keyword(s):  
Explicit Formula ◽  
The Other ◽  
Column Vector ◽  
Other Hand ◽  
Minimum Discrepancy ◽  
Positive Integers ◽  
Maximum Discrepancy ◽  
Sum Vector

Let $m$ and $n$ be positive integers, and let $R=(r_1,\ldots, r_m)$ and $ S=(s_1,\ldots, s_n)$ be non-negative integral vectors. Let ${\cal A} (R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$, and let $\bar A$ be the $m \times n$ $(0,1)$-matrix where for each $i$, $1\le i \le m$, row $i$ consists of $r_i$ $1$'s followed by $n-r_i$ $0$'s. If $S$ is monotone, the discrepancy $d(A)$ of $A$ is the number of positions in which $\bar A$ has a $1$ and $A$ has a $0$. It equals the number of $1$'s in $\bar A$ which have to be shifted in rows to obtain $A$. In this paper, we study the minimum and maximum $d(A)$ among all matrices $A \in {\cal A} (R,S)$. We completely solve the minimum discrepancy problem by giving an explicit formula in terms of $R$ and $S$ for it. On the other hand, the problem of finding an explicit formula for the maximum discrepancy turns out to be very difficult. Instead, we find an algorithm to compute the maximum discrepancy.

On the Wassermann Reaction, and especially its Significance in Relation to General Paralysis

1909 ◽  
Vol 55 (230) ◽  
pp. 437-447 ◽  
Author(s):  
Carl Hamilton Browning ◽  
Ivy McKenzie
Keyword(s):  
Congenital Syphilis ◽  
Silver Impregnation ◽  
The Other ◽  
Impregnation Method ◽  
Other Hand ◽  
Starting Point ◽  
Fresh State ◽  
Extreme Variability ◽  
Wassermann Reaction ◽  
Striking Fact

The discovery by Schaudinn of the Spirochæta pallida was the starting-point of an extensive series of investigations which have thrown much light on the nature of syphilitic infection and its consequences. Reliable authorities are agreed as to the aetiological relationship between this organism and syphilitic disease, and its presence can with ease be demonstrated in chancres, syphilides, and the tissues of cases of congenital syphilis. In tertiary lesions its presence has been noted in gummata and in aortitis, though only in a very few cases. In the so-called para-syphilitic diseases it has not yet been seen. If it be the case that the presence of the organism be indispensable to the production of tertiary and para-syphilitic lesions, the difficulty of demonstrating it may be due to one or both of the following causes: (1) The organisms may be present in another form representing a different stage in their life cycle; or (2) they may be so few in number as to render demonstration extremely difficult, as is the case with the tubercle bacillus in the lesions of chronic fibroid phthisis. The difficulty consequent on a paucity in numbers may be enhanced by difficulty in staining. Two methods of staining the spirochete are in use–Giemsa's stain for film preparations, and a silver impregnation method for the examination of tissues. It is a remarkable fact that tissue which shows enormous numbers of silver impregnated organisms may show very few or none at all in the films stained by Giemsa's method, while on the other hand we have noted an extreme variability in the extent to which impregnation by silver may be obtained; for example, if the tissues of a syphilitic infant be fixed soon after death and while still in a very fresh state, impregnation may be very slight in some parts and distinct in others; thus, in the case of a syphilitic pneumonia, the organisms in the fresh proliferating pneumonic tissue are not seen, or are represented by delicate spirals which are recognised with difficulty, while in the desquamating and degenerating epithelium of the bronchi they may be much more distinct. It is, on the other hand, a striking fact that the spirochaetes in a syphilitic fætus which has been dead for some time are usually impregnated easily with silver, and this is in accord with other observations in the use of the silver impregnation method in demonstrating structures.

scholarly journals $k$-Fold Sidon Sets

10.37236/3860 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Javier Cilleruelo ◽  
Craig Timmons
Keyword(s):  
The Other ◽  
Sidon Sets ◽  
Sidon Set ◽  
Other Hand ◽  
Positive Integers

Let $k \geq 1$ be an integer.  A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3 + c_4 = 0$.  We prove that for any integer $k \geq 1$, a $k$-fold Sidon set $A \subset [N]$ has at most $(N/k)^{1/2} + O((Nk)^{1/4})$ elements. Indeed we prove that given any $k$ positive integers $c_1<\cdots <c_k$, any set $A\subset [N]$ that contains only trivial solutions to $c_i(x_1-x_2)=c_j(x_3-x_4)$ for each $1 \le i \le j \le k$, has at most $(N/k)^{1/2}+O((c_k^2N/k)^{1/4})$ elements. On the other hand, for any $k \geq 2$ we can exhibit $k$ positive integers $c_1,\dots, c_k$ and a set $A\subset [N]$ with $|A|\ge (\frac 1k+o(1))N^{1/2}$, such that $A$ has only trivial solutions to $c_i(x_1 - x_2) = c_j (x_3 -  x_4)$ for each $1 \le i \le j\le k$.

scholarly journals The Order of Appearance of the Product of Five Consecutive Lucas Numbers

2014 ◽  
Vol 59 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Diego Marques ◽  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Fibonacci Number ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Positive Integers

Abstract Let Fn be the nth Fibonacci number and let Ln be the nth Lucas number. The order of appearance z(n) of a natural number n is defined as the smallest natural number k such that n divides Fk. For instance, z(Fn) = n = z(Ln)/2 for all n > 2. In this paper, among other things, we prove that for all positive integers n ≡ 0,8 (mod 12).

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