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ON THE DIOPHANTINE EQUATION (an)x + (bn)y = (cn)z

Mi-Mi Ma|Jian-Dong Wu
대한수학회 (The Korean Mathematical Society)|2015.07|대한수학회보 (Bulletin of the KMS)|제52권 제4호|pp. 1133-1138

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참고문헌

키워드

Diophantine equationPythagorean tripleJe´smanowicz’ conjecture
주제분류수학

초록

In 1956, Je´smanowicz conjectured that, for any positive integer n and any primitive Pythagorean triple (a, b, c) with a2+b2 = c2, the equation (an)x+(bn)y = (cn)z has the unique solution (x, y, z) = (2, 2, 2). In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples (a, b, c) = (4k2 − 1, 4k, 4k2 + 1).

Abstract

In 1956, Je´smanowicz conjectured that, for any positive integer n and any primitive Pythagorean triple (a, b, c) with a2+b2 = c2, the equation (an)x+(bn)y = (cn)z has the unique solution (x, y, z) = (2, 2, 2). In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples (a, b, c) = (4k2 − 1, 4k, 4k2 + 1).

참고문헌 (16)

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