Takakuwa, Kei; Asaeda, You On a conjecture on Pythagorean numbers. (English) Zbl 0796.11009 Proc. Japan Acad., Ser. A 69, No. 7, 252-255 (1993). L. Jésmanowicz conjectured that if \(u\), \(v\), \(w\) are Pythagorean numbers, then the diophantine equation on \(\ell,m,n\in\mathbb{N}\) \[ u^ \ell+ v^ m= w^ n \] has the unique solution \((\ell,m,n)= (2,2,2)\). In this paper, the authors prove this conjecture in some special cases. Reviewer: Ke Zhao (Chengdu) Cited in 2 ReviewsCited in 7 Documents MSC: 11D09 Quadratic and bilinear Diophantine equations Keywords:quadratic diophantine equation; Pythagorean numbers Citations:Zbl 0796.11010 PDFBibTeX XMLCite \textit{K. Takakuwa} and \textit{Y. Asaeda}, Proc. Japan Acad., Ser. A 69, No. 7, 252--255 (1993; Zbl 0796.11009) Full Text: DOI References: [1] L. Jesmanowicz: Kilka uwag o liczbach pitagorejwkich (Some remarks on Pythagorean numbers). Wiadom. Mat, 1, 196-202 (1956). · Zbl 0074.27205 [2] N. Terai: The diophantine equation x2 + qm = pn. Acat Arith., LXIII.4, 351-358 (1993). · Zbl 0770.11020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.