A note on Jeśmanowicz’ conjecture concerning Pythagorean numbers. (English) Zbl 0849.11036

Guo, Yongdong; Le, Maohua

Comment. Math. Univ. St. Pauli 44, No. 2, 225-228 (1995).

It has been conjectured by L. Jeśmanowicz [Wiadom. Mat. 1, 196-202 (1956; Zbl 0074.27205) (in Polish)] that the equation \[ (r^2- s^2)^x+ (2rs)^y= (r^2+ s^2)^z \quad \text{ for }r,s\in \mathbb{N},\;(r,s)=1,\;r>s,\;2\mid rs\tag{1} \] in positive integers \(x\), \(y\), \(z\) has only the solution given by \(x= y= z=2\). The authors obtain the following result: if \(2\parallel r\), \(r\geq 6000\) and \(s=3\) then equation (1) does not have any other solution than \((x, y, z)= (2, 2, 2)\).

Reviewer: R.J.Stroeker (Rotterdam)

MSC:

11D61	Exponential Diophantine equations
11D09	Quadratic and bilinear Diophantine equations

Keywords:

Jeśmanowicz’ conjecture; Pythagorean numbers; exponential diophantine equation

Citations:

Zbl 0074.27205