共查询到20条相似文献,搜索用时 327 毫秒
1.
Wolfgang Rump 《Archiv der Mathematik》2017,108(1):29-43
In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers \({B_n}\) are given by the recurrence \({B_n = 6 B_{n-1} - B_{n-2}}\) with initial conditions \({B_0 = 0, B_1 = 1}\) and its associated Lucas balancing numbers \({C_n}\) are given by the recurrence \({C_n = 6 C_{n-1} - C_{n-2}}\) with initial conditions \({C_0 = 1, C_1 = 3}\). First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (x, y) for the Diophantine equation \({2x^2 + 1 = 3^b y^m}\) for any even positive integers b and m with \({m > 2}\), given in (Int J Number Theory 11:1259–1274, 2015). Also we prove that the Diophantine equations \({B_n B_{n+d}\ldots B_{n+(k-1)d} = y^m}\) and \({C_n C_{n+d}\ldots C_{n+(k-1)d} = y^m}\) have no solution for any positive integers n, d, k, y, and m with \({m \geq 2, y \geq 2}\) and gcd\({(n,d) = 1}\). 相似文献
2.
Urban Larsson Neil A. McKay Richard J. Nowakowski Angela A. Siegel 《International Journal of Game Theory》2018,47(2):613-652
We introduce a class of normal-play partizan games, called Complementary Subtraction. These games are instances of Partizan Subtraction where we take any set A of positive integers to be Left’s subtraction set and let its complement be Right’s subtraction set. In wythoff partizan subtraction we take the set A and its complement B from wythoff nim, as the two subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we can find their binary representation in polynomial time using a representation of integers as sums of Fibonacci numbers, known as the ternary (or “the even”) Fibonacci representation. 相似文献
3.
Takafumi Miyazaki 《The Ramanujan Journal》2018,45(3):601-613
For any given odd prime p and a fixed positive integer D prime to p, we study the equation \(x^2+D^m=p^n\) in positive integers x, m and n. We use a classical work of Dem’janenko in 1965 on a certain quadratic Diophantine equation together with some results concerning the existence of primitive divisors of Lucas sequences to examine our equation when D is a product of \(p-1\) and a square. 相似文献
4.
Odd indexed Fibonacci numbers F2n+1 can be written as sums of two squares a 2+b 2. In this paper, we study the distribution of the lattice points (a,b) on the circles of radius \(\sqrt{F_{2n}+1}\). 相似文献
5.
Ali Reza Ashrafi Jernej Azarija Khadijeh Fathalikhani Sandi Klavžar Marko Petkovšek 《Annals of Combinatorics》2016,20(2):209-229
The Fibonacci cube \({\Gamma_{n}}\) is obtained from the n-cube Q n by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube \({\Lambda_{n}}\) is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of vertex orbit sizes of \({\Lambda_{n}}\) is \({\{k \geq 1; k |n\} \cup \{k \geq 18; k |2n\}}\), the number of vertex orbits of \({\Lambda_{n}}\) of size k, where k is odd and divides n, is equal to \({\sum_{d | k} \mu (\frac{k}{d})F_{\lfloor{\frac{d}{2}}\rfloor+2}}\), and the number of edge orbits of \({\Lambda_{n}}\) is equal to the number of vertex orbits of \({\Gamma_{n-3}}\). Dihedral transformations of strings and primitive strings are essential tools to prove these results. 相似文献
6.
Pavel Trojovský 《P-Adic Numbers, Ultrametric Analysis, and Applications》2017,9(3):228-235
Let (F n ) n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as . In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.
相似文献
$${\left[ {\begin{array}{*{20}{c}} m \\ k \end{array}} \right]_F} = \frac{{{F_{m - k + 1}} \cdots {F_{m - 1}}{F_m}}}{{{F_1} \cdots {F_k}}}$$
7.
Sandro Bettin Jonathan W. Bober Andrew R. Booker Brian Conrey Min Lee Giuseppe Molteni Thomas Oliver David J. Platt Raphael S. Steiner 《The Ramanujan Journal》2018,45(3):659-681
By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions. 相似文献
8.
Ick Sun Eum 《印度理论与应用数学杂志》2018,49(3):521-526
The Catalan-Mersenne numbers cn are double Mersenne numbers defined by c0 = 2 and \({c_n} = {2^{{c_{n - 1}}}} - 1\) for positive integers n. We prove a certain congruence relation of the Catalan- Mersenne numbers. 相似文献
9.
Peter Frankl 《Israel Journal of Mathematics》2017,222(1):421-430
Let s > k ≧ 2 be integers. It is shown that there is a positive real ε = ε(k) such that for all integers n satisfying (s + 1)k ≦ n < (s + 1)(k + ε) every k-graph on n vertices with no more than s pairwise disjoint edges has at most \(\left( {\begin{array}{*{20}{c}} {\left( {s + 1} \right)k - 1} \\ k \end{array}} \right)\) edges in total. This proves part of an old conjecture of Erd?s. 相似文献
10.
Let S be a subset of a finite abelian group G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay+(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z3 and Zn2 n, n ≥ 1, where Zk is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs. 相似文献
11.
MAHADI DDAMULIRA FLORIAN LUCA MIHAJA RAKOTOMALALA 《Proceedings Mathematical Sciences》2017,127(3):411-421
In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 2. 相似文献
12.
For a polynomial f of odd degree, nontrivial S-units can be effectively related to the continued fraction expansion of elements involving the square root of the polynomial f only in the case where S consists of an infinite valuation and a finite valuation determined by a first-degree polynomial h. In the paper, the proof that the quasi-periodicity of the continued fraction expansion of an element of the form \(\frac{{\sqrt f }}{{{h^s}}}\) implies periodicity is completed. In particular, it is proved that the continued fraction expansion of \(\sqrt f \) for f of any degree is quasi-periodic in k((h)) it and only if it is periodic. 相似文献
13.
Alexandra Shlapentokh 《Israel Journal of Mathematics》2018,226(2):579-633
We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold: (1) For any rational prime q and any positive rational integer m. algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by qm. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set \(\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P \ne q\) is any prime such that qm +1 ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions.We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable. 相似文献
14.
Piero Filipponi 《Rendiconti del Circolo Matematico di Palermo》1921,45(1):37-56
A particular use of well-known combinatorial expressions for Fibonacci and Lucas numbers gives rise to two interesting classes of integers (namely, the numbersF n(k) andL n(k)) governed by the integral parametersn andk. After establishing the main properties of these numbers and their interrelationship, we study some congruence properties ofL n(k), one of which leads to a supposedly new characterisation of prime numbers. A glimpse of possible generalisations and further avenues of research is also caught. 相似文献
15.
V. P. Kostov 《Functional Analysis and Its Applications》2009,43(2):147-150
Every real polynomial of degree n in one variable with root ?1 can be represented as the Schur-Szeg? composition of n ? 1 polynomials of the form (x + 1) n?1(x + a i ), where the numbers a i are uniquely determined up to permutation. Some a i are real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ? ρ, r ? [n/2], there exists a polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a i . 相似文献
16.
Sh. T. Ishmukhametov B. G. Mubarakov Kamal Al-Anni Maad 《Russian Mathematics (Iz VUZ)》2017,61(11):26-33
Bezout’s equation is a representation of the greatest common divisor d of integers A and B as a linear combination Ax + By = d, where x and y are integers called Bezout’s coefficients. The task of finding Bezout’s coefficients has numerous applications in the number theory and cryptography, for example, for calculation of multiplicative inverse elements in modular arithmetic. Usually Bezout’s coefficients are caclulated using the extended version of the classical Euclidian algorithm.We elaborate a new algorithm for calculating Bezout’s coefficients based on the k-ary GCD algorithm. 相似文献
17.
We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the ‘normal’ statistics given by the Gauss-Kuzmin measure. As a byproduct, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer k and a quadratic irrational α, the length of the period of the continued fraction expansion of k n α equals ck n + o(k15n/16) for some positive constant c. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces. 相似文献
18.
On the basis of a given criterion for the quasi-periodicity of continued fractions for elements of the hyperelliptic field L = K(x)(\(\sqrt f \)), where K is an arbitrary field of characteristic different from 2 and f ∈ K[x] is a square-free polynomial, new polynomials f ∈ Q[x] of odd degree for which the elements of \(\sqrt f \) have periodic continued fraction expansion are found. 相似文献
19.
B. M. Phong 《Acta Mathematica Hungarica》2016,148(2):450-465
We give a characterization of those multiplicative functions f and g for which \({f(p +m^2) = g(p) + g(m^2)}\) and \({g(p^2) = g(p)^2}\) are satisfied for all primes p and all positive integers m. 相似文献
20.
Liangang Ma Radhakrishnan Nair 《Proceedings of the Steklov Institute of Mathematics》2017,299(1):157-177
Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi’s backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number x, if (p n /q n )n≥1 denotes its sequence of regular continued fraction convergents, set θ n (x) = q n 2 |x ? p n /q n |, n = 1, 2.... The metric behaviour of the Cesàro averages of the sequence (θ n (x))n≥1 has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence (θ n (x))n≥1 for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of \(({\theta _{{k_n}}}(x))\)n≥1 for certain sequences (k n )n≥1, initiated by the second named author, to Haas–Molnar maps. 相似文献