1,121 research outputs found
Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields
For each positive integer it is shown how to construct a finite
collection of multivariable polynomials such that each positive integer whose squareroot has
a continued fraction expansion with period lies in the range of exactly
one of these polynomials. Moreover, each of these polynomials satisfy a
polynomial Pell's equation (where
and are polynomials in the variables ) and the fundamental solution can be written down.
Likewise, if all the 's and are non-negative then the continued
fraction expansion of can be written down. Furthermore, the
congruence class modulo 4 of depends in a simple way on the variables
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 do there exist positive
integers and such that .Comment: 13 page
The irrationality of a number theoretical series
Denote by the sum of the -th powers of the divisors of ,
and let . We prove that Schinzel's
conjecture H implies that is irrational, and give an unconditional proof
for the case
- β¦