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On the integer solutions of exponential equations in function fields

Umberto Zannier (2004)

Annales de l’institut Fourier

This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal...

On the intersection of two distinct k -generalized Fibonacci sequences

Diego Marques (2012)

Mathematica Bohemica

Let k 2 and define F ( k ) : = ( F n ( k ) ) n 0 , the k -generalized Fibonacci sequence whose terms satisfy the recurrence relation F n ( k ) = F n - 1 ( k ) + F n - 2 ( k ) + + F n - k ( k ) , with initial conditions 0 , 0 , , 0 , 1 ( k terms) and such that the first nonzero term is F 1 ( k ) = 1 . The sequences F : = F ( 2 ) and T : = F ( 3 ) are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation F n ( k ) = F m ( ) . In this note, we use transcendental tools to provide a general method for finding the intersections F ( k ) F ( m ) which gives evidence supporting...

On the Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

We completely solve the Diophantine equations x ² + 2 a q b = y (for q = 17, 29, 41). We also determine all C = p a p k a k and C = 2 a p a p k a k , where p , . . . , p k are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) i = 1 n X i - d i = 1 n Y ² i = - I with d ∈ N for nonsingular X i , Y i M ( Z ) , i=1,...,n.

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