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12
Sigma function solution of the initial value problem for Somos 5 sequences
, 2008
"... The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we ..."
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Cited by 14 (8 self)
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The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.
On the generation of heronian triangles
"... ABSTRACT. We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O `n2+ε´. We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems. 1. ..."
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Cited by 5 (3 self)
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ABSTRACT. We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O `n2+ε´. We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems. 1.
Heron Triangles And Elliptic Curves
 Bull. Aust. Math. Soc
, 1998
"... In this paper we present a proof that there exist innitely many rational sided triangles with two rational medians and rational area. These triangles correspond to rational points on an elliptic curve of rank one. We also display three triangles (one previously unpublished) which do not belong to an ..."
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Cited by 3 (1 self)
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In this paper we present a proof that there exist innitely many rational sided triangles with two rational medians and rational area. These triangles correspond to rational points on an elliptic curve of rank one. We also display three triangles (one previously unpublished) which do not belong to any of the known innite families. I.
Symmetry, Integrability and Geometry: Methods and Applications Quantum SuperIntegrable Systems as Exactly Solvable Models ⋆
"... Abstract. We consider some examples of quantum superintegrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between superintegrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Fi ..."
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Abstract. We consider some examples of quantum superintegrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between superintegrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic algebras are thus constructed in a way analogous to that of the highest weight representations of Lie algebras. Key words: quantum integrability; superintegrability; exact solvability; Laplace–Beltrami
ELLIPTIC CURVES COMING FROM HERON TRIANGLES
"... Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v 2 = u(u − a)(u − b)(u − c). The Heron formula states that Q = √ P (P − a)(P − b)(P − c) where P is the semiperimeter of the triangle, so the point (u, v) = (P ..."
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Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v 2 = u(u − a)(u − b)(u − c). The Heron formula states that Q = √ P (P − a)(P − b)(P − c) where P is the semiperimeter of the triangle, so the point (u, v) = (P, Q) is a rational point on the quartic. Also the point of in nity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curve y 2 = (x + a b)(x + b c)(x + c a). The point (P, Q) on the quartic transforms to −2abc (x, y) =
Symmetries of triangles with two rational medians ∗
, 2003
"... We study the symmetry group of the solutions to equations which define rational triangles with two rational medians. The group action is used to discover three new elliptic curves parametrizing such triangles which also have rational area. We prove that only finitely many of the rational triangles w ..."
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We study the symmetry group of the solutions to equations which define rational triangles with two rational medians. The group action is used to discover three new elliptic curves parametrizing such triangles which also have rational area. We prove that only finitely many of the rational triangles with two rational medians and rational area, which correspond to rational points on eight elliptic curves, can also have a third rational median. Finally, we present a new analysis of sporadic examples of such triangles along with the discovery of a new sporadic triangle.
ELLIPTIC CURVES AND TRIANGLES WITH THREE RATIONAL MEDIANS
"... Abstract. In his paper Triangles with three rational medians, about the characterization of all rationalsided triangles with three rational medians, Buchholz proves that each such triangle corresponds to a point on a oneparameter family of elliptic curves whose rank is at least 2. We prove that in ..."
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Abstract. In his paper Triangles with three rational medians, about the characterization of all rationalsided triangles with three rational medians, Buchholz proves that each such triangle corresponds to a point on a oneparameter family of elliptic curves whose rank is at least 2. We prove that in fact the exact rank of the family in Buchholz paper is 3. We also exhibit a subfamily whose rank is at least 4 and we prove the existence of in nitely many curves of rank 5 over Q parametrized by an elliptic curve of positive rank. Finally, we show particular examples of curves within those families having rank 9 and 10 over Q. 1.