A remark on a theorem of Serret

Abstract

Suppose f(x, y) is an indefinite quadratic form with integer coefficients and irrational roots. Let μ(f) denote the minimum value of |f(x, y)| on Z × Z{minus 45 degree rule}{(0,0)}. The author shows that for any solution (x, y) ∈ Z × Z of f(x, y) = ±μ(f) the ratio x y is a convergent of the continued fraction of one of the roots of f, except in the case where f(x, y) is equivalent but not equal to the form x2 - xy - y2. In this latter case there is only one exceptional solution, which the author completely describes. This improves a special case of a theorem of Serret [8]. © 1986.