Książka - szczegóły

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 $C_{p}: y² = Φ_{p}(x)$ and $D_{p}: py² = Φ_{p}(x)$, where $Φ_{p}$ is the pth cyclotomic polynomial, and we determine the sets $C_{p}(ℚ)$ and $D_{p}(ℚ)$. Using the elliptic Chabauty method, we prove that $C_{p}(ℚ) = {(-1,-1),(-1,1),(0,-1),(0,1)}$ and $D_{p}(ℚ) = {(1,-1),(1,1)}$ for p ∈ {11,13,17}.