On the multiplicity in Pillai's problem with Fibonacci numbers and powers of a fixed prime

Batte, Herbert; Ddamulira, Mahadi; Kasozi, Juma; Luca, Florian

doi:10.3336/gm.57.2.02

Cited by 3 publications

(2 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…to allow to vary the recurrence sequences (U n ) n∈N and (V n ) n∈N in the result of Chim, Pink and Ziegler [3]. In particular, Batte, Ddamulira, Kasozi and Luca [1] showed that for all pairs (p, c) ∈ P × Z with p a prime, the Diophantine equation…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Pillai’s Problem involving Lucas sequences of the second kind

Heintze,

Ziegler

2024

Res. number theory

View full text Add to dashboard Cite

In this paper, we consider the Diophantine equation $$ V_n - b^m = c $$ V n - b m = c for given integers b, c with $$ b \ge 2 $$ b ≥ 2 , whereas $$ V_n $$ V n varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (n, m) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of $$ V_n $$ V n .

show abstract

Section: Introductionmentioning

confidence: 99%

“…Furthermore let κ = 1 if K is real and κ = 2 otherwise. For all integers j with 1 ≤ j ≤ N choose A j ≥ max{Dh(η j ), | log η j |, 0.16},1 This prime p is called a primitive divisor.…”

mentioning

confidence: 99%

On Pillai’s Problem involving Lucas sequences of the second kind

Heintze,

Ziegler

2024

Res. number theory

View full text Add to dashboard Cite

show abstract

Solutions to a Pillai-Type Equation Involving Tribonacci Numbers and S-Units

Batte,

Luca

2024

Mediterr. J. Math.

View full text Add to dashboard Cite

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

et al. 2023

View full text Add to dashboard Cite

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.

show abstract

On the multiplicity in Pillai's problem with Fibonacci numbers and powers of a fixed prime

Abstract: Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) be a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for the number of distinct representations of \(c\) as \(F_k-p^\ell\) with \(k\ge 2\) and \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).

Cited by 3 publications

References 11 publications

On Pillai’s Problem involving Lucas sequences of the second kind

On Pillai’s Problem involving Lucas sequences of the second kind

Solutions to a Pillai-Type Equation Involving Tribonacci Numbers and S-Units

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

Contact Info

Product

Resources

About