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

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  • Nai-juan Deng
  • Ridi Huang

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.

Suggested Citation

  • Nai-juan Deng & Ridi Huang, 2026. "Diophantine Equation 4k2−1nx+4kny=4k2+1nz," Journal of Mathematics, Hindawi, vol. 2026, pages 1-5, March.
    • Handle: RePEc:hin:jjmath:3566998
    DOI: 10.1155/jom/3566998
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