# Sur les courbes hyperelliptiques cyclotomiques et les équations ${x}^{p}-{y}^{p}=cz²$

• 2007

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## Abstract

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Let p be a prime number ≥ 11 and c be a square-free integer ≥ 3. In this paper, we study the diophantine equation ${x}^{p}-{y}^{p}=cz²$ in the case where p belongs to 11,13,17. More precisely, we prove that for those primes, there is no integer solution (x,y,z) to this equation satisfying gcd(x,y,z) = 1 and xyz ≠ 0 if the integer c has the following property: if ℓ is a prime number dividing c then ℓ ≢ 1 mod p. To obtain this result, we consider the hyperelliptic curves ${C}_{p}:y²={\Phi }_{p}\left(x\right)$ and ${D}_{p}:py²={\Phi }_{p}\left(x\right)$, where ${\Phi }_{p}$ is the pth cyclotomic polynomial, and we determine the sets ${C}_{p}\left(ℚ\right)$ and ${D}_{p}\left(ℚ\right)$. Using the elliptic Chabauty method, we prove that ${C}_{p}\left(ℚ\right)=\left(-1,-1\right),\left(-1,1\right),\left(0,-1\right),\left(0,1\right)$ and ${D}_{p}\left(ℚ\right)=\left(1,-1\right),\left(1,1\right)$ for p ∈ 11,13,17.

## How to cite

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Wilfrid Ivorra. Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$. 2007. <http://eudml.org/doc/286030>.

@book{WilfridIvorra2007,
author = {Wilfrid Ivorra},
keywords = {higher degree Diophantine equations; Chabauty's method; -rational points; hyperelliptic curves; cyclotomic polynomial},
language = {fre},
title = {Sur les courbes hyperelliptiques cyclotomiques et les équations $x^\{p\} - y^\{p\} = cz²$},
url = {http://eudml.org/doc/286030},
year = {2007},
}

TY - BOOK
AU - Wilfrid Ivorra
TI - Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$
PY - 2007
LA - fre
KW - higher degree Diophantine equations; Chabauty's method; -rational points; hyperelliptic curves; cyclotomic polynomial
UR - http://eudml.org/doc/286030
ER -

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