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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.

The circle method and pairs of quadratic forms

Henryk Iwaniec, Ritabrata Munshi (2010)

Journal de Théorie des Nombres de Bordeaux

We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

The method of infinite ascent applied on A 4 ± n B 3 = C 2

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

Each of the Diophantine equations A 4 ± n B 3 = C 2 has an infinite number of integral solutions ( A , B , C ) for any positive integer n . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A , B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions ( A , B , C ) of the Diophantine equation a A 3 + c B 3 = C 2 for any co-prime integer pair ( a , c ) .

Variations on a theme of Runge: effective determination of integral points on certain varieties

Aaron Levin (2008)

Journal de Théorie des Nombres de Bordeaux

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our...

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