42 research outputs found

    On the periodic writing of cubic irrationals and a generalization of RĂ©dei functions

    Get PDF
    In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.</jats:p

    Kuroda's formula and arithmetic statistics

    Full text link
    Kuroda's formula relates the class number of a multi-quadratic number field KK to the class numbers of its quadratic subfields kik_i. A key component in this formula is the unit group index Q(K)=[OK×:∏iOki×]Q(K) = [\mathcal{O}_{K}^{\times}: \prod_i\mathcal{O}_{k_i}^{\times}]. We study how Q(K)Q(K) behaves on average in certain natural families of totally real biquadratic fields KK parametrized by prime numbers

    On the Diophantine equation x2+q2m=2ypx^2+q^{2m}=2y^p

    Full text link
    In this paper we consider the Diophantine equation x2+q2m=2ypx^2+q^{2m}=2y^p where m,p,q,x,ym,p,q,x,y are integer unknowns with m>0,m>0, pp and qq are odd primes and gcd⁥(x,y)=1.\gcd(x,y)=1. We prove that there are only finitely many solutions (m,p,q,x,y)(m,p,q,x,y) for which yy is not a sum of two consecutive squares. We also study the above equation with fixed yy and with fixed $q.

    Primitive divisors of Lucas and Lehmer sequences

    Get PDF
    Stewart reduced the problem of determining all Lucas and Lehmer sequences whose nn-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for n≀30n \leq 30. Further computations lead us to conjecture that, for n>30n > 30, the nn-th element of such sequences always has a primitive divisor

    Lucas sequences whose 12th or 9th term is a square

    Full text link
    Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_n(P,Q) is defined by U_0=0, U_1=1, U_n = P U_{n-1}-Q U_{n-2} for n>1. The sequence {U_n(1,-1)} is the familiar Fibonacci sequence, and it was proved by Cohn that the only perfect square greater than 1 in this sequence is U12=144U_{12}=144. The question arises, for which parameters P, Q, can U_n(P,Q) be a perfect square? In this paper, we complete recent results of Ribenboim and MacDaniel. Under the only restriction GCD(P,Q)=1 we determine all Lucas sequences {U_n(P,Q)} with U_{12}= square. It turns out that the Fibonacci sequence provides the only example. Moreover, we also determine all Lucas sequences {U_n(P,Q) with U_9= square.Comment: 13 page

    There are no multiply-perfect Fibonacci numbers

    Get PDF
    Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors
    corecore