Diophantine Equation 4k2−1nx+4kny=4k2+1nz

Nai-juan Deng and Ridi Huang

Journal of Mathematics, 2026, vol. 2026, 1-5

Abstract: Let a,b,c be a primitive Pythagorean triple such that a2+b2=c2 with 2b. In 1956, L. Jesmanowicz conjectured that, for any positive integer n, the equation anx+bny=cnz has only the positive solution x,y,z=2,2,2. In 1959, Lu Wenduan claimed that if n=1 and a,b,c=4k2−1,4k,4k2+1, then the conjecture is true. Denote by Pn the product of the prime factors of n. In this paper, we prove that the conjecture is true for n>1,a,b,c=4k2−1,4k,4k2+1, under some conditions.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:3566998

DOI: 10.1155/jom/3566998

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