A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples. II.
The exponential equation \(a^x+b^y=c^z\) in positive integers \(x,y,z\), where \(a,b,c\) are integers larger than \(1\), has attracted ample attention. Recently, \textit{R. Scott} and \textit{R. Styer} [Publ. Math. 88, No. 1--2, 131--138 (2016; Zbl 1374.11057)] proved that such an equation cannot more than \(2\) solutions if \(c\) is odd. In the paper under review, the authors consider the special case that \((a,b,c)\) is a primitive Pythagorean triple, i.e., \(a^2+b^2=c^2\) and \(\gcd (a,b,c)=1\). Then we already have a solution \((x,y,z)=(2,2,2)\), and \textit{L. Jeśmanowicz} [Ann. Soc. Math. Pol., Ser. II, Wiad. Mat. 1, No. 2, 196--202 (1956; Zbl 0074.27205)] conjectured that in this case there are no other solutions. This conjecture is still open, but it has been proved in various special cases. The authors prove Jeśmanowicz' conjecture in a couple of other cases. Recall that if \((a,b,c)\) is a primitive Pythagorean triple, and \(b\) is even, then there are coprime positive integers \(m,n\) such that \(a=m^2-n^2\), \(b=2mn\), and \(c=m^2+n^2\). So we consider the equation (*) \((m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z\) in positive integers \(x,y,z\). Using quartic residue symbols over the Gaussian integers and elementary congruence considerations, the authors prove Jeśmanowicz' conjecture for pairs \((m,n)\) in various residue classes modulo \(4\), \(8\) and \(16\). As a consequence they deduce that if \(n\equiv 2\, ({\bmod}\, 4)\) then (*) has no solutions with \(y>1\) other than \((2,2,2)\). The study of solutions with \(y=1\) turned out to be more complicated. Using Laurent's lower bound [Acta Arith. 133, No. 4, 325--348 (2008; Zbl 1215.11074)] on linear forms in two logarithms, together with computer calculations, the authors show that (*) has no solutions with \(y=1\) if \(n\equiv 2\, ({\bmod}\, 4)\) and \(n<600\). In both these results, no restriction has to be imposed on \(m\).
scientific article; zbMATH DE number 6837294
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