Displaying similar documents to “Perfect powers expressible as sums of two fifth or seventh powers”

On the equation a³ + b³ⁿ = c²

Michael A. Bennett, Imin Chen, Sander R. Dahmen, Soroosh Yazdani (2014)

Acta Arithmetica

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We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.

Arithmetic of elliptic curves and diophantine equations

Loïc Merel (1999)

Journal de théorie des nombres de Bordeaux

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We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms.

Modular parametrizations of certain elliptic curves

Matija Kazalicki, Koji Tasaka (2014)

Acta Arithmetica

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Kaneko and Sakai (2013) recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and Ramanujan-Serre differential operator. In this paper, we study certain properties of the modular parametrization associated to the elliptic curves over ℚ, and as a consequence we generalize and explain some of their findings. ...

Primitive Points on a Modular Hyperbola

Igor E. Shparlinski (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.