1 citations · 1 across the 4 of their papers we have counts for
6 papers
Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory
Barry R. Smith
Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form t…
New algorithms for modular inversion and representation by binary quadratic forms arising from structure in the Euclidean algorithm
Christina Doran, Shen Lu, Barry R. Smith
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for com…
Reducing quadratic forms by kneading sequences
Barry R. Smith
We introduce an invertible operation on finite sequences of positive integers and call it "kneading". Kneading preserves three invariants of sequences -- the parity of the length,…
End-symmetric continued fractions and quadratic congruences
Barry R. Smith
We show that for a fixed integer , the congruence has the solution with if and only if has a continued fraction e…
The remainders of the Euclidean algorithm with symmetric quotients
Barry R. Smith
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in…
Divisibility of partial zeta function values at zero for degree 2p extensions
Barry Smith
Let K/k be an Abelian extension of number fields, S be a set of places of k, and p be an odd prime number. We continue an earlier investigation of the author into the values at zer…