A note on a conjecture of Jeśmanowicz

Deng, Moujie; Cohen, G.

Deng, Moujie ; Cohen, G.

Colloquium Mathematicae, Tome 84/85 (2000), p. 25-30 / Harvested from The Polish Digital Mathematics Library

Résumé

Let a, b, c be relatively prime positive integers such that $a^{2} + b^{2} = c^{2}$ . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ${(a n)}^{x} + {(b n)}^{y} = {(c n)}^{z}$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${(u^{2} - v^{2})}^{x} + {(2 u v)}^{y} = {(u^{2} + v^{2})}^{z}$ , for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.

Publié le : 2000-01-01

Zbl 0960.11026

EUDML-ID : urn:eudml:doc:210838

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     author = {Moujie Deng and G. Cohen},
     title = {A note on a conjecture of Je\'smanowicz},
     journal = {Colloquium Mathematicae},
     volume = {84/85},
     year = {2000},
     pages = {25-30},
     zbl = {0960.11026},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv86i1p25bwm}
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Deng, Moujie; Cohen, G. A note on a conjecture of Jeśmanowicz. Colloquium Mathematicae, Tome 84/85 (2000) pp. 25-30. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv86i1p25bwm/

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