On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Ddamulira, Mahadi; Luca, Florian

doi:10.1007/s11139-020-00278-7

Cited by 4 publications

(40 citation statements)

References 10 publications

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“…This is Subsection 2.2 and 2.3 in [1]. All the results in these two Subsections 2.2 and 2.3 of [1] are well stated and explained.…”

Section: Linear Forms In Logarithms and Continued Fractionssupporting

confidence: 63%

“…Subsection 2.1 in [1] was correctly written. All results there-in are correct and therefore we also adopt this subsection as it appears from page 652 to page 654 of [1].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [1], it was shown that there exists no positive integer solutions (r , n, m, x) of the Diophantine equation (1.3) with r ≥ 3 and x = 2. However, this came with a big computational mistake in stating and applying Theorem 2.1 in [4].…”

Section: Introductionmentioning

confidence: 99%

“…However, this came with a big computational mistake in stating and applying Theorem 2.1 in [4]. In this paper, we restudy [1]. Still, we consider equation (1.3) in nonnegative integers (r , n, m, x) treating r ≥ 3 as an integer parameter, since the cases r ∈ {1, 2} were treated in [3] and [6] respectively.…”

Section: Introductionmentioning

confidence: 99%

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On the exponential diophantine equation $$U_{n}^x+U_{n+1}^x=U_m$$

Batte,

Ddamulira,

Kasozi

et al. 2024

Ramanujan J

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Let $$ \{U_n\}_{n\ge 0} $$ { U n } n ≥ 0 be the Lucas sequence. For integers x, n and m, we find all solutions to $$U_{n}^x+U_{n+1}^x=U_m$$ U n x + U n + 1 x = U m . The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms. http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf, 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem 2.1 from Mignotte (A kit on linear forms in three logarithms. http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf, 2008).

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“…This is Subsection 2.2 and 2.3 in [1]. All the results in these two Subsections 2.2 and 2.3 of [1] are well stated and explained.…”

Section: Linear Forms In Logarithms and Continued Fractionssupporting

confidence: 63%

“…Subsection 2.1 in [1] was correctly written. All results there-in are correct and therefore we also adopt this subsection as it appears from page 652 to page 654 of [1].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the exponential diophantine equation $$U_{n}^x+U_{n+1}^x=U_m$$

Batte,

Ddamulira,

Kasozi

et al. 2024

Ramanujan J

Self Cite

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“…has yielded very rich results, but some important problems about it are far from being solved (see [7]). In recent 20 years, many authors have considered equation (1) when A, B and C are Fibonacci A00045 or Lucas A000204 or Pell A000129 numbers in OEIS [14] or when A and B are Fibonacci or Lucas or Pell numbers (see [1,2,4,6,12,13,18]).…”

Section: Introductionmentioning

confidence: 99%

A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

Mao-hua

Soydan

2020

Glas. Mat. Ser. III

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Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

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On the Diophantine Equation

HASANALIZADE

2024

Bull. Aust. Math. Soc.

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A generalisation of the well-known Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$ , $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $n\ge 0$ is the k-generalised Pell sequence $\{P^{(k)}_n\}_{n\ge -(k-2)}$ whose first k terms are $0,\ldots ,0,1$ and each term afterwards is given by the linear recurrence $P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+\cdots +P^{(k)}_{n-k}$ . For the Pell sequence, the formula $P^2_n+P^2_{n+1}=P_{2n+1}$ holds for all $n\ge 0$ . In this paper, we prove that the Diophantine equation $$ \begin{align*} (P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m \end{align*} $$ has no solution in positive integers $k, m$ and n with $n>1$ and $k\ge 3$ .

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On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Abstract: Let r ≥ 1 be an integer and U := (U n) n≥0 be the Lucas sequence given by U 0 = 0, U 1 = 1, and U n+2 = rU n+1 + U n , for all n ≥ 0. In this paper, we show that there are no positive integers r ≥ 3, x = 2, n ≥ 1 such that U x n + U x n+1 is a member of U.

Cited by 4 publications

References 10 publications

On the exponential diophantine equation $$U_{n}^x+U_{n+1}^x=U_m$$

On the exponential diophantine equation $$U_{n}^x+U_{n+1}^x=U_m$$

A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

On the Diophantine Equation

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