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Eisenstein's proof of the Quadratic Reciprocity (QR) (and its Jacobi symbol generalization) both rely on counting lattice points in two congruent triangles. If we take $t$-dilates of these triangles, does an extension of Gauss' Lemma still hold with some dependence on $t$? That is, can we assign a Legendre symbol $(p|q)$, or some simple extension thereof depending on $t$, to $(-1)^{M(t)}$, where $M(t)$ is the number of lattice points in the $t$-dilate of the same triangle used in the standard proof and derive from it a "$t$-dilated" version of QR?

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what is a t-dilate? – quanta Mar 11 '11 at 17:20
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Suppose $P$ is an $n$-polytope of the form $\text{conv}\{\mathbf{0}, a_1 \mathbf{e}_1, \dots, a_n \mathbf{e}_n \}$. Then a non-trivial $t$-dilate $t P$ is the convex hull $\text{conv}\{\mathbf{0}, a_1 t \mathbf{e}_1, \dots, a_n t \mathbf{e}_n \}$, where $t > 1$. The triangles above correspond to taking $n = 2$. – user02138 Mar 11 '11 at 17:33

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