Query: Pythagorean Triples exponents

title: Pythagorean Triples with Common Sides
abstract: There exist a finite number of Pythagorean triples that have a common leg. In this paper we derive the formulas that generate pairs of primitive Pythagorean triples with common legs and also show the process of how to determine all the primitive and nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.

title: New Formula to Generate All Pythagorean Triples with Proof and Geometrical Interpretation
abstract: This way has the convenience to find easily all Pythagorean triples x, y, z N, where x is a predetermined integer, which means finding all right triangles whose sides have integer measures and one cathetus is predetermined.

title: An open problem on Jesmanowicz' conjecture concerning primitive Pythagorean triples
abstract: Let m 31 be an even integer with gcd(m, 31) 1. In this paper, using some elementary methods, we prove that the equation (m2 312)x (62m) (m2 +312)z has only the positive integer solution (x, y, z) (2, 2, 2) This result resolves an open problem raised by T. Miyazaki (Acta Arith. 186 (2018) 1 36) about Jesmanowicz' conjecture concerning primitive Pythagorean triples.

title: Two unique primitive pythagorean triples from every integer
abstract: In this paper we show that for every integer, there are two unique primitive solutions to the classical Pythagorean equation. These solutions have two interesting points. The two unique primitive triples correspond to every integer, with no additional conditions. Two, for each of the solutions, C B is the set of all odd integers squared.

title: On Matrix Pythagorean Triples
abstract: We present a parametrization for the Pythagorean matrices, i.e. integer valued matrices A, B, and C such that

title: Direct parametrization of Pythagorean triples
abstract: If the two axes of symmetry of a quadratic form in two variables have integer coefficients, the reflection across the axes defines a group action on the primitive solutions of the Diophantine equation defined by the quadratic form. In this paper, we introduce quadratic forms with rational axes of symmetry that admit a single set of polynomials which parametrize their primitive solutions up to the reflections.

title: Formally Verifying the Solution to the Boolean Pythagorean Triples Problem
abstract: The Boolean Pythagorean Triples problem asks: does there exist a binary coloring of the natural numbers such that every Pythagorean triple contains an element of each color? This problem was first solved in 2016, when Heule, Kullmann and Marek encoded a finite restriction of this problem as a propositional formula and showed its unsatisfiability. In this work we formalize their development in the theorem prover Coq. We state the Boolean Pythagorean Triples problem in Coq, define its encoding as a propositional formula and establish the relation between solutions to the problem and satisfying assignments to the formula. We verify Heule et al.'s proof by showing that the symmetry breaks they introduced to simplify the propositional formula are sound, and by implementing a correct by construction checker for proofs of unsatisfiability based on reverse unit propagation.

title: Some new families of positive rank elliptic curves arising from Pythagorean triples
abstract: In the present paper, we introduce some new families of elliptic curves with positive rank arising from Pythagorean triples. We study elliptic curves of the form y x Ax+B, where A,B {a, b, c} are two different numbers and (a, b, c) is a rational Pythagorean triple. First of all, we prove that if (a, b, c) is a primitive Pythagorean triple (PPT) then the rank of each family is positive. Furthermore, we construct subfamilies of rank at least 3 in each family but one with rank at least 2, and obtain elliptic curves of high rank in each family. Finally, we consider two other new families of elliptic curves of the forms y x(x a)(x c) and y x(x b)(x+c) and prove that if (a, b, c) is a PPT, then the rank of each family is positive.

title: A NOTE ON THE GENERATION OF PYTHAGOREAN TRIPLES
abstract: We give a new method (up to our knowledge) for generating some primitive pythagorean triples. Our method is based on the construction of a new primitive pythagore an triple from a predetermined primitive pythagorean triple. We generalize this idea and we construct a new pythagorean triple from a predetermined pythagorean n tuple.

title: Spherical Pythagorean triples, and volume rationality of spherical tetrahedra
abstract: We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive examples to this conjecture. As a by product, we also obtain a classification of all spherical Pythagorean triples, previously found by Smith.