42 research outputs found
On the periodic writing of cubic irrationals and a generalization of RĂ©dei functions
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
Kuroda's formula relates the class number of a multi-quadratic number field
to the class numbers of its quadratic subfields . A key component in
this formula is the unit group index . We study how behaves on average in
certain natural families of totally real biquadratic fields parametrized by
prime numbers
On the Diophantine equation
In this paper we consider the Diophantine equation where
are integer unknowns with and are odd primes and
We prove that there are only finitely many solutions
for which is not a sum of two consecutive squares. We also
study the above equation with fixed and with fixed $q.
Primitive divisors of Lucas and Lehmer sequences
Stewart reduced the problem of determining all Lucas and Lehmer sequences
whose -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 . Further computations
lead us to conjecture that, for , the -th element of such sequences
always has a primitive divisor
Lucas sequences whose 12th or 9th term is a square
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
. 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
Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors