# On the Diophantine equation $ax^{2t}+bx^ty+cy^2=d$ and pure powers in recurrence sequences.

@article{Shorey1983OnTD, title={On the Diophantine equation \$ax^\{2t\}+bx^ty+cy^2=d\$ and pure powers in recurrence sequences.}, author={T. N. Shorey and C. L. Stewart}, journal={Mathematica Scandinavica}, year={1983}, volume={52}, pages={24-36} }

## 85 Citations

Sums of $S$-units and perfect powers in recurrence sequences

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Let S := {p1, . . . , pl} be a finite set of primes and denote by US the set of all rational integers whose prime factors are all in S. Let (Un)n≥0 be a non-degenerate linear recurrence sequence with…

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Perfect powers in linear recurring sequences

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Let α1, . . . , αt be the distinct roots of the corresponding characteristic polynomial X − A1Xk−1 − . . .− Ak. (2) Then for n ≥ 0, Gn = P1(n)α 1 + P2(n)α n 2 + . . .+ Pt(n)α n t , where Pi(n) is a…

Yet another generalization of Sylvester's theorem and its application

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In this paper, we consider Sylvester’s theorem on the largest prime divisor of a product of consecutive terms of an arithmetic progression, and prove another generalization of this theorem. As an…

ON POLYNOMIAL VALUES OF THE PRODUCT OF THE TERMS OF LINEAR RECURRENCE SEQUENCES

- Mathematics
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where G0, G1, . . . , Gk−1, A1, A2, . . . , Ak are rational integer constants. We need an other sequence, too. Let H = {Hn} ∞ n=0 be another linear recurrence of order l defined by Hn = B1Hn−1 + · ·…

RESULTS CONCERNING PRODUCTS AND SUMS OF THE TERMS OF LINEAR RECURRENCES

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Many papers have investigated perfect powers and polynomial values as terms of linear recursive sequences of rational integers. Many results show, under some restrictions, that if a term of a…

On Fibonacci powers

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Fibonacci numbers have engaged the attention of mathematicians for several centuries, and whilst many of their properties are easy to establish by very simple methods, there are several unsolved…

Dedicated to my friend Marco Fontanu on his 50th birthday

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We investigate when certain sums or differences of terms of Lucas sequences are powers (or multiples of powers). In the case of squares, we obtain explicit bounds, explicit solution in the case of…

Pure powers in recurrence sequences

- Mathematics
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Let G be a linear recursive sequence of order k satisfying the recursion Gn=A1Gn 1+��� +AkGn k. In the case k=2 it is known that there are only finitely many perfect powers in such a sequence.…