On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

Erazo, Harold S.; Gómez, Carlos A.; Luca, Florian

doi:10.1142/s1793042121500871

Cited by 3 publications

(4 citation statements)

References 14 publications

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“…Using Inequality (13) as well as h(b) = log b gives us Comparing this lower bound with the upper bound (14) gives us…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In the classical setting, it is possible to "unfix" a and b completely: Bennett [3] proved that for any integers a, b ≥ 2 and c ≥ 1 Equation (1) has at most two solutions (n, m) ∈ Z 2 >0 . Moreover, he conjectured that in fact the Erazo, Gómez, Luca [14] k-Fibonacci numbers Pell numbers Bravo, Díaz, Gómez [4] Table 1. Overview of results on U n − V m = c equation has at most one solution (n, m) for all but 11 specific exceptional triples (a, b, c).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Diophantine equation $U_n-b^m = c$

Heintze¹,

Tichy²,

Vukusic³

et al. 2022

Preprint

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Let (Un) n∈N be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B and N 0 such that for any b, c ∈ Z with b > B the equation Un − b m = c has at most two distinct solutions (n, m) ∈ N 2 with n ≥ N 0 and m ≥ 1. Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1, T 3 = 2 and Tn = T n−1 + T n−2 + T n−3 for n ≥ 4. By means of the LLL-algorithm and continued fraction reduction we are able to prove N 0 = 1.1 • 10 37 and B = e 438 . The corresponding reduction algorithm is implemented in Sage.

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“…Using Inequality (13) as well as h(b) = log b gives us Comparing this lower bound with the upper bound (14) gives us…”

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Diophantine equation $U_n-b^m = c$

Heintze¹,

Tichy²,

Vukusic³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…yields a refinement of (9). Let S(P) be the set of solutions of (10) which are not solutions of ( 10…”

Section: The Proof Of Theorem 21mentioning

confidence: 99%

“…For related result, for example concerning sums or linear combinations of integers with fixed prime factors in the solution sets of Pell equations, see e.g. the papers [7,9,18] and the references there.…”

Section: Introductionmentioning

confidence: 99%

Terms of recurrence sequences in the solution sets of generalized Pell equations

Hajdu

Sebestyén

2022

Int. J. Number Theory

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On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐

et al. 2023

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Let ( U n ) n ∈ N (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B B and N 0 N_0 such that for any b , c ∈ Z b,c\in \mathbb {Z} with b > B b> B the equation U n − b m = c U_n - b^m = c has at most two distinct solutions ( n , m ) ∈ N 2 (n,m)\in \mathbb {N}^2 with n ≥ N 0 n\geq N_0 and m ≥ 1 m\geq 1 . Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1 T_1= T_2=1 , T 3 = 2 T_3=2 and T n = T n − 1 + T n − 2 + T n − 3 T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n ≥ 4 n\geq 4 . By means of the LLL-algorithm and continued fraction reduction we are able to prove N 0 = 2 N_0=2 and B = e 438 B=e^{438} . The corresponding reduction algorithm is implemented in Sage.

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On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

Cited by 3 publications

References 14 publications

On the Diophantine equation $U_n-b^m = c$

On the Diophantine equation $U_n-b^m = c$

Terms of recurrence sequences in the solution sets of generalized Pell equations

On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐

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