A note on the shuffle variant of Jeśmanowicz’ conjecture. (English) Zbl 1390.11071

Let \((a,b,c)\) be a primitive Pythagorean triple. According to the conjecture of Jeśmanowicz, in this case the only solution of the equation \(a^x+b^y=c^z\) in positive integers \(x,y,z\) is given by \((x,y,z)=(2,2,2)\). If here we also have \(c=b+1\), then \(a^2=b+c\). Miyazaki conjectured that the only solution to the equation \(c^x+b^y=a^z\) in positive integers \(x,y,z\) is \((x,y,z)=(1,1,2)\) if \(c=b+1\), and the equation has no solutions with \(c>b+1\). He proved the conjecture if \(c\equiv 1 \pmod{b}\). In the present paper the author extends this result by showing that the conjecture of Miyazaki is true if \(c\equiv 1 \pmod{b/2^{\text{ord}_2(b)}}\).


11D61 Exponential Diophantine equations
Full Text: DOI Euclid