On the Equations zm = F (x, y ) and Axp + Byq = Czr

From MaRDI portal
Publication:4863835

DOI10.1112/blms/27.6.513zbMath0838.11023OpenAlexW1967973885WikidataQ56610650 ScholiaQ56610650MaRDI QIDQ4863835

Henri Darmon, Andrew Granville

Publication date: 10 June 1996

Published in: Bulletin of the London Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1112/blms/27.6.513




Related Items

Orbifolds, special varieties and classification theory.On $abc$ and discriminantsA REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x2 + y2n = z3On the Diophantine equation \(x^ 2+a^ 2=2y^ p\).Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactesVanishing Polynomial SumsA Local Version of Szpiro's ConjectureLecture on the abc Conjecture and Some of Its ConsequencesArithmetic progressions consisting of unlike powersOn integers which are not differences of two powersAround the Chevalley-Weil theoremIntegral points and orbits of endomorphisms on the projective planeA modular approach to the generalized Ramanujan-Nagell equationLocally nilpotent derivations of rings with roots adjoinedThe equation \(x^{2n}+y^{2n}=z^5\)Twists of \(X(7)\) and primitive solutions to \(x^2+y^3=z^7\)Fermat-type equations of signature \((13,13,p)\) via Hilbert cuspformsOn squares in Lucas sequencesGeneralized Fermat equations: A miscellanyThe local-global principle for integral points on stacky curvesThe canonical ring of a stacky curveExceptional cases of Terai's conjecture on Diophantine equationsA new parametrization for ideal classes in rings defined by binary forms, and applicationsDiophantine equations with three monomialsShimura curves and the abc conjectureAsymptotic Fermat for signatures \((r,r,p)\) using the modular approachElliptic curves over the rationals with good reduction outside two odd primesℚ-Curves, Hecke characters and some Diophantine equationsSolving 𝑆-unit, Mordell, Thue, Thue–Mahler and Generalized Ramanujan–Nagell Equations via the Shimura–Taniyama ConjectureAsymptotic Fermat for signature \((4, 2, p)\) over number fields\(\mathbb{Q}\)-curves, Hecke characters, and some Diophantine equations. IIMost odd-degree binary forms fail to primitively represent a squareKlein forms and the generalized superelliptic equationAN APPLICATION OF THE MODULAR METHOD AND THE SYMPLECTIC ARGUMENT TO A LEBESGUE–NAGELL EQUATIONPerfect powers generated by the twisted Fermat cubicReverse engineered Diophantine equationsA multi-Frey approach to Fermat equations of signature $(r,r,p)$Jeśmanowicz' conjecture on exponential Diophantine equationsChen's conjecture and its generalizationSolving \(a x^p + b y^p = c z^p\) with \(abc\) containing an arbitrary number of prime factorsOn some ternary Diophantine equations of signature \((p,p,k)\)NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5AThe Diophantine equation \((x^{k} - 1)(y^{k} - 1) = (z^{k} - 1)^{t}\)The Diophantine equation \(Ax^ p+By^ q=Cz^ r\).Arithmetic Aspects of Orbifold PairsQuantum \((abc)\)-theoremsINTEGERS REPRESENTED BY REVISITEDUnnamed ItemTHE DIOPHANTINE EQUATION A4 + 2δB2 = CnA note on the high power Diophantine equationsOn the Diophantine system \(a^2+ b^2 = c^3\) and \(a^x + b^y = c^z\) for \(b\) is an odd primeAurifeuillian factorizationCubic forms, powers of primes and classification of elliptic curvesOn powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjectureFiniteness results for \(F\)-Diophantine setsPerfect power Riesel numbersA new generalization of Fermat's last theoremHomogeneous forms of degree 3 and \(p\)-th powersPolynomial values of products of terms from an arithmetic progressionOn \(px^{2}+q^{2n}=y^{p}\) and related Diophantine equationsPolynomial parametrization for the solutions of Diophantine equations and arithmetic groupsGalois theory, discriminants and torsion subgroup of elliptic curvesOn the irreducibility of the iterates of 𝑥ⁿ-𝑏Équations de Fermat de type (5, 5,p)Solutions of some generalized Ramanujan-Nagell equationsAn elementary proof of Fermat's last theorem for all even exponentsCovers of the integers with odd moduli and their applications to the forms $x^{m}-2^{n}$ and $x^{2}-F_{3n}/2$On the equation $s^2+y^{2p} = \alpha^3$Sums of two cubes as twisted perfect powers, revisitedTERAI'S CONJECTURE ON EXPONENTIAL DIOPHANTINE EQUATIONSAffine cones over smooth cubic surfacesThe generalised Fermat equation \(x^{2} + y^{3} = z^{15}\)Perfect powers expressible as sums of two cubesThe generalized Fermat equation with exponents 2, 3,Solutions of the Diophantine equation \(7X^2 + Y^7 = Z^2\) from recurrence sequencesShift operators defined in the Riordan group and their applicationsPowerful arithmetic progressionsOn the greatest prime factor of sides of a Heron triangleInteger solutions to the equation \(y^2=x(x^2\pm p^k)\)Arithmetic of elliptic curves and diophantine equationsOn ternary Diophantine equations of signature(p,p,2)over number fieldsLinearly dependent powers of binary quadratic formsA ``class group obstruction for the equation \(Cy^d=F(x,z)\)Solving Fermat-type equations via modular ℚ-curves over polyquadratic fieldsEqual sums of two cubes of quadratic formsThe primitive solutions to \(x^3+y^9=z^2\)Enumerative Galois theory for cubics and quarticsArithmetic progressions of squares, cubes and \(n\)-th powersCanonical Heights and Preperiodic Points for Certain Weighted Homogeneous Families of PolynomialsMiyanishi's characterization of the affine 3-space does not hold in higher dimensionsRecipes to Fermat-type equations of the form \(x^r + y^r =Cz^p\)Sur l'équation a3+ b3= cpThe Darmon-Granville equation with algebraic exponents.