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Siegel’s theorem and the Shafarevich conjecture

Aaron Levin (2012)

Journal de Théorie des Nombres de Bordeaux

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k , one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S . We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus g would imply an effective version of Siegel’s theorem for integral points on...

Solutions entières de l’équation Y m = f ( X )

Dimitrios Poulakis (1991)

Journal de théorie des nombres de Bordeaux

Soit K un corps de nombres. Dans ce travail nous calculons des majorants effectifs pour la taille des solutions en entiers algébriques de K des équations, Y 2 = f ( X ) , où f ( X ) ∈ K [ X ] a au moins trois racines d’ordre impair, et Y m = f ( X ) où m ≥ 3 et f ( X ) ∈ K [ X ] a au moins deux racines d’ordre premier à m . On améliore ainsi les estimations connues ([2],[9]) pour les solutions de ces équations en entiers algébriques de K .

Solutions of the Diophantine Equation 7 X 2 + Y 7 = Z 2 from Recurrence Sequences

Hayder R. Hashim (2020)

Communications in Mathematics

Consider the system x 2 - a y 2 = b , P ( x , y ) = z 2 , where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7 X 2 + Y 7 = Z 2 if ( X , Y ) = ( L n , F n ) (or ( X , Y ) = ( F n , L n ) ) where { F n } and { L n } represent the sequences of Fibonacci numbers and Lucas numbers respectively....

Some observations on the Diophantine equation f(x)f(y) = f(z)²

Yong Zhang (2016)

Colloquium Mathematicae

Let f ∈ ℚ [X] be a polynomial without multiple roots and with deg(f) ≥ 2. We give conditions for f(X) = AX² + BX + C such that the Diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial integer solutions and prove that this equation has a rational parametric solution for infinitely many irreducible cubic polynomials. Moreover, we consider f(x)f(y) = f(z)² for quartic polynomials.

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