Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita; Takafumi Miyazaki

Colloquium Mathematicae, Tome 126 (2012), p. 211-222 / Harvested from The Polish Digital Mathematics Library

Résumé

Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b \equiv 0 (m o d 2^{r})$ and $b \equiv \pm 2^{r} (m o d a)$ for some non-negative integer r, then the Diophantine equation $a^{x} + b^{y} = c^{z}$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).

Publié le : 2012-01-01

Zbl 1291.11071

EUDML-ID : urn:eudml:doc:284026

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     author = {Yasutsugu Fujita and Takafumi Miyazaki},
     title = {Je\'smanowicz' conjecture with congruence relations},
     journal = {Colloquium Mathematicae},
     volume = {126},
     year = {2012},
     pages = {211-222},
     zbl = {1291.11071},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-2-6}
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Yasutsugu Fujita; Takafumi Miyazaki. Jeśmanowicz' conjecture with congruence relations. Colloquium Mathematicae, Tome 126 (2012) pp. 211-222. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-2-6/