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Articles 31 - 33 of 33
Full-Text Articles in Discrete Mathematics and Combinatorics
Multi-Variable Polynomial Solutions To Pell's Equation And Fundamental Units In Real Quadratic Fields, James Mclaughlin
Multi-Variable Polynomial Solutions To Pell's Equation And Fundamental Units In Real Quadratic Fields, James Mclaughlin
Mathematics Faculty Publications
Solving Pell’s equation is of relevance in finding fundamental units in real quadratic fields and for this reason polynomial solutions are of interest in that they can supply the fundamental units in infinite families of such fields. In this paper an algorithm is described which allows one to construct, for each positive integer n, a finite collection, {Fi}, of multi-variable polynomials (with integral coefficients), each satisfying a multi-variable polynomial Pell’s equation C 2 i − FiH 2 i = (−1)n−1 , where Ci and Hi are multi-variable polynomials with integral coefficients. Each positive integer whose square-root has a regular continued …
On The Divergence In The General Sense Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin
On The Divergence In The General Sense Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin
Mathematics Faculty Publications
We show, for each q-continued fraction G(q) in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which G(q) diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other q-continued fractions, for example the G¨ollnitz-Gordon continued fraction, on the unit circle.
Polynomial Solutions To Pell's Equation And Fundamental Units In Real Quadratic Fields, James Mclaughlin
Polynomial Solutions To Pell's Equation And Fundamental Units In Real Quadratic Fields, James Mclaughlin
Mathematics Faculty Publications
Finding polynomial solutions to Pell’s equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. In this paper, for each triple of positive integers (c, h, f) satisfying c 2 − f h2 = 1, where (c, h) are the smallest pair of integers satisfying this equation, several sets of polynomials (c(t), h(t), f(t)) which satisfy c(t) 2 − f(t) h(t) 2 = 1 and (c(0), h(0), f(0)) = (c, h, f) are derived. Moreover, it is shown that the pair (c(t), h(t)) constitute the fundamental polynomial …