Linear combinations of prime powers in X -coordinates of Pell equations
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Linear combinations of prime powers in X-coordinates of Pell equations Harold Erazo1 · Carlos A. Gómez1
· Florian Luca2,3,4
Received: 22 September 2018 / Accepted: 25 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let {X }≥1 be the sequence of X -coordinates of the positive integer solutions (X , Y ) of the Pell equation X 2 − dY 2 = ±1 corresponding to a nonsquare integer d > 1. We show that there are only a finite number of nonsquare integers d > 1 such that there are at least two different elements of the sequence {X }≥1 that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents. Moreover, we solve explicitly the case in which two of the X -coordinates above are a sum of a power of two and a power of three under the above condition on the exponents. This work is motivated by the recent paper Bertók et al. (Int J Number Theory 13(02):261–271, 2017). Keywords Pell equations · Linear combinations of prime powers · Lower bounds for linear forms in logarithms Mathematics Subject Classification 11B39 · 11D45 · 11J86
F. L. was supported in parts by Grant CPRR160325161141 of NRF and the Number Theory Focus Area Grant of CoEMaSS at Wits (South Africa) and CGA 17-02804S (Czech Republic).
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Carlos A. Gómez [email protected] Harold Erazo [email protected] Florian Luca [email protected]
1
Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100–00, Cali, Colombia
2
School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, South Africa
3
Department of Mathematics, Faculty of Sciences, University of Ostrava, 30 Dubna 22, 701 03 Ostrava 1, Czech Republic
4
Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia
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H. Erazo et al.
1 Introduction For a nonsquare integer d > 1 and the Pell equation X 2 − dY 2 = ε ∈ {±1},
where
X , Y ∈ Z+ ,
(1)
it is well-known that all its positive integer solutions (X , Y ) have the form √ √ √ X + Y d = X k + Yk d = (X 1 + Y1 d)k for some k ∈ Z+ , where (X 1 , Y1 ) is the smallest positive integer solution of (1). It is convenient to set X 0 = 1. The sequence {X k }k≥1 is a binary recurrent sequence, satisfying the recurrence relation X k = 2X 1 X k−1 − ε X k−2 ,
for all k ≥ 2,
(2)
where ε is chosen according to (1). Furthermore, the Binet formula √ √ (X 1 + Y1 d)k + (X 1 − Y1 d)k Xk = 2
(3)
holds for all nonnegative integers k. There has been recent interest around investigating for which d, there are at least two members of the sequence {X k }k≥1 which belong to some interesting sequence of positive integers. For example, for some sequences it is known that there is an integer D such that if d ≥ D then at most one member of the sequence {X k }k≥1 can belong to such sequences. Indeed, in [16] it is shown that X k is a Fibonacci number for at most one k,
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