2008 | OriginalPaper | Chapter
On the Diophantine Equation G n (x) = G m (y) with Q (x, y)=0
Authors : Clemens Fuchs, Attila Pethő, Robert F. Tichy
Published in: Diophantine Approximation
Publisher: Springer Vienna
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by (Link opens in a new window)
Let
K
denote an algebraically closed field of characteristic 0, and let A
0
,..., A
d–1
, G
0
,...,
G
d-
1
∈ K[
X
] and
$$ \left( {Gn\left( X \right)} \right)_{n = 0}^\infty $$
be a sequence of polynomials defined by the
d-
th order linear recurring relation
(1)
$$ G_{n + d} \left( X \right) = A_{d - 1} \left( X \right)G_{n + d - 1} \left( X \right) + \cdots + A_0 \left( X \right)G_n \left( X \right), for n \geqslant 0. $$
Furthermore, let P(
X
) ∈ K[
X
], deg
P
≥ 1. Recently, we investigated the question, what can be said about the number of solutions of the Diophantine equation
(2)
$$ Gn\left( X \right) = Gm\left( {P\left( X \right)} \right). $$