On Rational Cubic Residues
نویسنده
چکیده
In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M/L ≡ (t − 1)/(t − 9t) mod q for some integer t. Recently, Z. Sun has stated a similar result for cubic nonresidues which follows from several corollaries appearing in an earlier paper of his. In this paper we provide an independent, self-contained development of an equivalent result. In particular, we describe a family gγ(t) of rational functions involving cubic polynomials that play an analogous role to the function (t − 1)/(t − 9t) appearing in Lehmer’s theorem. keywords: Cubic residue, cubic reciprocity MSC 2000: 11A15
منابع مشابه
Convex Surface Visualization Using Rational Bi- cubic Function
The rational cubic function with three parameters has been extended to rational bi-cubic function to visualize the shape of regular convex surface data. The rational bi-cubic function involves six parameters in each rectangular patch. Data dependent constraints are derived on four of these parameters to visualize the shape of convex surface data while other two are free to refine the shape of s...
متن کاملAn Optimal G^2-Hermite Interpolation by Rational Cubic Bézier Curves
In this paper, we study a geometric G^2 Hermite interpolation by planar rational cubic Bézier curves. Two data points, two tangent vectors and two signed curvatures interpolated per each rational segment. We give the necessary and the sufficient intrinsic geometric conditions for two C^2 parametric curves to be connected with G2 continuity. Locally, the free parameters w...
متن کاملA Clean Approach to Rational Cubic Residues
In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result essentially states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M L ≡ t 2 − 1 t3 − 9t mod q for some integer t. In this paper we demonstrate that a similar result may be obtained for cubic nonresidues, yielding a cubic character for fixed p that...
متن کاملShape Preserving Interpolation Using C2 Rational Cubic Spline
Abstract: This study proposes new C rational cubic spline interpolant of the form cubic/quadratic with three shape parameters to preserves the geometric properties of the given data sets. Sufficient conditions for the positivity and data constrained modeling of the rational interpolant are derived on one parameter while the remaining two parameters can further be utilized to change and modify t...
متن کاملLooking for Rational Curves on Cubic Hypersurfaces
The aim of these lectures is to study rational points and rational curves on varieties, mainly over finite fields Fq. We concentrate on hypersurfaces Xn of degree ≤ n+ 1 in Pn+1, especially on cubic hypersurfaces. The theorem of Chevalley–Warning (cf. Esnault’s lectures) guarantees rational points on low degree hypersurfaces over finite fields. That is, if X ⊂ Pn+1 is a hypersurface of degree ≤...
متن کامل