Mou-Jie Deng, Jin Guo: Application of quartic residue character theory to the Diophantine equation $a^x + b^y = c^z$, 133-139

Abstract:

Let $(a, b, c)$ be a primitive Pythagorean triple satisfying $a^2 +b^2 = c^2.$ In 1956, Jesmanowicz conjectured that the only positive integer solution of the exponential Diophantine equation $a^x + b^y = c^z$ is $x = y = z = 2.$ For the primitive Pythagorean triple $(a, b, c)= (m^2 - n^2, 2mn, m^2 + n^2)$ with $m,n$ positive integers such that gcd $(m,n)=1$, $m>n, m\not\equiv n\pmod{2}$, many special cases of Jesmanowicz' conjecture have been settled under the condition that $2\parallel mn$. In this paper, using the theory of quartic residue character and elementary method together with some of Miyazaki's results, we show that Jesmanowicz' conjecture is true if $m\equiv4\pmod{8}, n\equiv9\pmod{16}$ or $m\equiv9\pmod{16}, n\equiv4\pmod{8}$.

Key Words: Exponential Diophantine equation, Pythagorean triple, Quartic residue character, Jesmanowicz' conjecture.

2010 Mathematics Subject Classification: Primary: 11D61.