Parity considerations in the expansion of Fermat-Pell polynomials
|
Descargar SCORM
Este recurso ha sido solicitado 2 veces (0 veces en los últimos 31 días).
Para poder solicitar este recurso debe identificarse como usuario de la biblioteca
|
| |
Ver
Detalles del recurso
|
|
|
Parity considerations in the expansion of Fermat-Pell polynomials
|
| Id. |
20814361 |
| Titulo |
Parity considerations in the expansion of Fermat-Pell polynomials |
| Autor(es) |
Laughlin, James Mc
|
| Localización |
http://arxiv.org/abs/math/0001190
|
| Versión |
1.0 |
| Estado |
Final
|
| Descripción |
For each positive integer $n$ it is shown how to construct a finite
collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor
\frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has
a continued fraction expansion with period $n+1$ lies in the range of exactly
one of these polynomials. Moreover, each of these polynomials satisfy a
polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where
$C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\lfloor
\frac{n+1}{2} \rfloor}$) and the fundamental solution can be written down.
Likewise, if all the $X_{i}$'s and $t$ are non-negative then the continued
fraction expansion of $\sqrt{F_{i}}$ can be written down. Furthermore, the
congruence class modulo 4 of $F_{i}$ depends in a simple way on the variables
$t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$ so that the fundamental unit
can be written down for a large class of real quadratic fields.
Along the way a complete solution is given to the problem of determining for
which symmetric strings of positive integers $a_{1},..., a_{n}$ do there exist
positive integers $D$ and $a_{0}$ such that $\sqrt{D} = [ a_{0};\bar{a_{1},
>..., a_{n},2a_{0}}]$.
|
| Palabras clave |
Mathematics - Number Theory
|
| Tipo de recurso |
Texto Narrativo
|
| Tipo de Interactividad |
Expositivo
|
| Nivel de Interactividad |
muy bajo
|
| Audiencia |
Estudiante
Profesor
Autor
|
| Estructura |
Atomic |
| Coste |
no
|
| Copyright |
sí
|
| Requerimientos técnicos |
Browser: Any
|
| Fecha de contribución |
26-mar-2007 |
| Contacto |
|
|
|
|
|
Valoración de los usuarios
No hay ninguna valoración para este recurso. Sea el primero en
valorar este recurso.
|
|
|
|
