Perfect powers in second order linear recurrences

<p>Let f(n)=1 if n=1, 2^(2^(n-2)) if n \\in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \\in {6,7,8,...}. We conjecture that if a system T \\subseteq {x_i+1=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that the conjecture and Matiyasevich's conjecture on finite-fold Diophantine representations are jointly inconsistent.</p>

Pure powers in recurrence sequences and some related diophantine equations

Let N = pq be an RSA modulus and e be a public exponent. Let j(N) = (p − 1)(q − 1) be the Euler’s totient function. The equation ex2 −j(N)y2 = z has infinitely many solutions in integers (x, y, z). We show that if x, y and z are suitably small, then one can factor the RSA modulus. Our bounds on the size of the solutions x, y, and z improve the existing bounds of some attacks on RSA such as Wiener’s continued fractions based attack, and Blömer-May’s lattice reduction based attack.

We find 48 more solutions to a Diophantine equation investigated by Martin Davis. Before our work, only two solutions were known. Construction of the new solutions required the factorization of several large integers. Because the equation relates to Hilbert’s Tenth Problem it is desirable to know if it has only finitely many solutions. An elaborate argument is given for the conjecture that the equation has infinitely many solutions in integers.

has only finitely many solutions in integers x and yv. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the assumption that the discriminant D(F) of F is nonzero. For most integers h these bounds improve upon those obtained by Bombieri and Schmidt in [5]. In the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of Nagell [30], Ore [32], Sandor [33], and Huxley [19]. In fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is, in general, best possible. In the second part we shall address the problem of finding forms F for which (1) has many solutions for arbitrarily large integers h. Finally we shall obtain upper bounds for the number of solutions of certain Thue-Mahler and Ramanujan-Nagell equations by appealing to estimates of Evertse, Gy6ry, Stewart, and Tijdeman [17] for the number of solutions of S-unit equations.

Let $p$ be a prime number with $p\neq 2$. We consider sequences generated by $n$th order linear recurrence relations over the finite field $Z_p$. In the first part of this paper we generalize some of the ideas in [6] to $n$th order linear recurrences. We then consider the case where the characteristic polynomial of the recurrence has one root in $Z_p$ of multiplicity $n$. In this case, we show that the corresponding recurrence can be generated by a relatively simple matrix.

The solutions of second order linear recurrences with periodical coefficients are shown to obey Benford’s law.

Objectives: The purpose of this research article is to find non-zero distinct integer solutions to the system of triple equations with five unknowns represented by . Different sets of distinct integer solutions are presented. Methods: Various substitution strategies and factorization techniques are employed for finding non-zero distinct integer solutions. Findings: Four different choices of substitutions are utilized to determine many non-zero distinct integer solutions to the given system of triple equations with five unknowns. Novelty: In this analysis, the given system of triple equations with five unknowns is reduced to a single non-homogeneous cubic equation for which solutions can be found elegantly. Keywords: System of triple equations, Triple equations with five unknowns, Non-homogeneous cubic Equation, Ternary cubic equation, Integer solutions

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Part I: Invited lectures. Domain Decomposition Methods M. Sarkis. Modification and Maintenance of ULV Decompositions J.L. Barlow. Advances in Jacobi Methods Z. Drmac, V. Hari, I. Slapnicar. Modelling of curved rods M. Jurak, J. Tambaca, Z. Tutek. Incompressible Newtonian flow through thin pipes E. Marusic-Paloka. Nonlinear Problems in Dynamics by the Finite Element in Time Method N. Kranjcevic, M. Stegic, N. Vrankovic. First Order Eigenvalue Perturbation Theory and the Newton Diagram J. Moro, F.M. Dopi. Part II: Contributed lectures. Microlocal energy density for hyperbolic systems N. Antonic, M. Lazar. Approximate solutions to some second order linear recurrences K. Balla, V. Horvat. Asymptotic Behaviour of Tension Spline Collocation Matrix I. Beros, M. Marusic. Numerical stability of Krylov subspace iterative methods for solving linear systems N. Bosner. Nonlinear Problems in Dynamics by the Finite Element in Time Method N. Kranjcevic, M. Stegic, N. Vrankovic. On directional bias of the Lp-norm T. Marosevic. A note on slip condition on corrugated boundary E. Marusic-Paloka. Relaxation of some energy functionals related to the formation of microstructure A. Raguz. A Coarse Space for Elasticity M. Sarkis. Numerical Approximations of the Sediment Transport Equations L. Sopta, N. Crnjaric-Zic, S. Vukovic. A model of irregular curved rods J. Tambaca. Existence of the density of states for some alloy type models with single site potentials that change sign I. Veselic. On principal eigenvalue of stationary diffusion problem with nonsymmetric coefficients M. Vrdoljak. Qualitative Analysis of some Solutions of Quasilinear System of Differential Equations B. Vrdoljak, A. Omerspahi . High-Order ENO and WENO Schemes with Flux Gradient and Source Term Balancing S. Vukovi , L. Sopta. Index.

We shall consider two curious divisibility properties due to (1, 5). Our main purpose is to generalize these properties for a general second order linear recursion. We use generating matrix approach for our purposes. By using our results, we derive a new recursive identities for two general second order linear recurrences. We give a more general result extending earlier divisibility properties on the similar subject. 2000 Mathematics Subject Classification: 11B37; 15B36

Sine sequence of second order linear recurrences

Completely p-primitive binary quadratic forms

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.

The binary quadratic Diophantine equation represented by is considered and analyzed for its non-zero distinct integer solutions for the choices of t given by (i) \(t=1\) (ii) \(t=3\) (iii) \(t=5\) (iv) \(t=2k\) and (v) \(t=2k+5\). A few interesting relations among the solutions are presented. Further, recurrence relations on the solutions are obtained. Keywords : Pell equation, integer solutions of Pell equation, binary quadratic Diophantine equation.