Zhang Xinwen and Zhang Wenpeng: The Exponential Diophantine Equation $((2^{2m}-1)n)^x+(2^{m+1}n)^y=((2^{2m}+1)n)^z$, p.337-344

Abstract:

Let $m,\ n$ be positive integers. Let $(a, b,
c)$ be a primitive Pythagorean triplet with $a^2+b^2=c^2$. In 1956, L. Jesmanowicz conjectured that the equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,
y, z)=(2, 2, 2)$. In this paper, using certain elementary methods, we prove that if $(a, b, c)=(2^{2m}-1, 2^{m+1}, 2^{2m}+1)$, then the above equation has only the positive integer solution $(x,
y, z)=(2, 2, 2)$. Thus it can be seen that Jesmanowicz's conjecture is true for infinitely many primitive Pythagorean triplets.

Key Words: Exponential diophantine equation, primitive Pythagorean triplet, Jesmanowicz's conjecture.

2000 Mathematics Subject Classification: Primary: 11D20;
Secondary: 11D61.

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