Geodesic
Jeśmanowicz' conjecture with congruence relations
Yasutsugu Fujita1 ; Takafumi Miyazaki2
1 Department of Mathematics College of Industrial Technology Nihon University 2-11-1 Shin-ei Narashino, Chiba, Japan
2 Department of Mathematics and Information Sciences Tokyo Metropolitan University 1-1 Minami-Ohsawa Hachioji, Tokyo, Japan
Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 211-222 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $a,b$ and $c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}$. We prove that if $b \equiv 0 \pmod{2^{r}}$ and $b \equiv \pm 2^{r} \pmod{a}$ 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 $c \equiv -1 \pmod{a}$.
DOI : 10.4064/cm128-2-6
Keywords: relatively prime positive integers prove equiv pmod equiv pmod non negative integer diophantine equation has only positive solution holds equiv pmod

Yasutsugu Fujita 1 ; Takafumi Miyazaki 2

1 Department of Mathematics College of Industrial Technology Nihon University 2-11-1 Shin-ei Narashino, Chiba, Japan
2 Department of Mathematics and Information Sciences Tokyo Metropolitan University 1-1 Minami-Ohsawa Hachioji, Tokyo, Japan 
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Yasutsugu Fujita; Takafumi Miyazaki. Jeśmanowicz' conjecture with congruence relations. Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 211-222. doi: 10.4064/cm128-2-6

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