A Geometrial Approach to the Diophantine Equation x_1^2+x_2^2+...+x_n^2=u^2
Abstract
We find all Diophantine solutions for the equation by refining the geometrical approach from (Ayoub, 1984) to find solutions of the equation We can find all rational points on the unit n-sphere by lines connecting those rational points to the point Such linear parametric equations will always have rational slopes. Keywords : Diophantine solutions ; Pythagorean triples ; unitsphereReferences
Ayoub, A. B. (1984). Integral Solutions to the Equation Mathematics Magazine, 57(4),
222-223.
Catalan, E. (1885). Bull. Acad. Roy. Belgique, 9(3), 531.
Cossali, P. (1797). Origine. Transporto in Italia. Algebra., 1, 97.
Dainelli, U. (1877). Giornale di Mat., 15, 378.
Gill, C. (1826). The Gentleman's Math. Companion, 29(5), 364.
Dickson, L. E. (1966). History of the Theory of Numbers (Vol. 2), Chelsea Publishing.
Euler, J. A. (1779). Acta Acad. Petrop., 3, 40.
Huerlimann, W. (2002). The Primitive Cuboids with Natural Edges and Diagonals According to Catalan and
Sierpinski. Mathematics Preprint Archive, 2002 (2), 444-452. (SSRN: https://ssrn.com/abstract=3133707)
Huerlimann, W. (2015). Hopf's quadratic map and permutation invariant properties of primitive cuboids,
Algebra Lett., 2015, Article ID 2.
Lebesgue, V. A. (1874). Nouv. Ann. Math., 13(2), 64.
Mikami, Y. (1912). Abh. Geschichte Math. Wiss., 30, 233.
222-223.
Catalan, E. (1885). Bull. Acad. Roy. Belgique, 9(3), 531.
Cossali, P. (1797). Origine. Transporto in Italia. Algebra., 1, 97.
Dainelli, U. (1877). Giornale di Mat., 15, 378.
Gill, C. (1826). The Gentleman's Math. Companion, 29(5), 364.
Dickson, L. E. (1966). History of the Theory of Numbers (Vol. 2), Chelsea Publishing.
Euler, J. A. (1779). Acta Acad. Petrop., 3, 40.
Huerlimann, W. (2002). The Primitive Cuboids with Natural Edges and Diagonals According to Catalan and
Sierpinski. Mathematics Preprint Archive, 2002 (2), 444-452. (SSRN: https://ssrn.com/abstract=3133707)
Huerlimann, W. (2015). Hopf's quadratic map and permutation invariant properties of primitive cuboids,
Algebra Lett., 2015, Article ID 2.
Lebesgue, V. A. (1874). Nouv. Ann. Math., 13(2), 64.
Mikami, Y. (1912). Abh. Geschichte Math. Wiss., 30, 233.
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Published
2021-09-06
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