17 research outputs found

    On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2

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    For an integer k2 k\geq 2 , let {Fn(k)}n0 \{F^{(k)}_{n} \}_{n\geq 0} be the k k--generalized Fibonacci sequence which starts with 0,,0,1 0, \ldots, 0, 1 (k k terms) and each term afterwards is the sum of the kk preceding terms. In this paper, we find all integers cc having at least two presentations as a difference between a kk--generalized Fibonacci number and a powers of 2 for any fixed k4k \geqslant 4. This paper extends previous work from [9] for the case k=2k=2 and [6] for the case k=3k=3

    On the Diophantine equation Unbm=cU_n-b^m = c

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    Let (Un)nN(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 BB and N0N_0 such that for any b,cZb,c\in \mathbb{Z} with b>Bb> B the equation Unbm=cU_n - b^m = c has at most two distinct solutions (n,m)N2(n,m)\in \mathbb{N}^2 with nN0n\geq N_0 and m1m\geq 1. Moreover, we apply our result to the special case of Tribonacci numbers given by T1=T2=1T_1= T_2=1, T3=2T_3=2 and Tn=Tn1+Tn2+Tn3T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n4n\geq 4. By means of the LLL-algorithm and continued fraction reduction we are able to prove N0=1.11037N_0=1.1\cdot 10^{37} and B=e438B=e^{438}. The corresponding reduction algorithm is implemented in Sage.Comment: 34 page

    On Pillai’s problem with Pell numbers and powers of 2

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    In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2

    On the Problem of Pillai with Fibonacci numbers and powers of 33

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    International audienceConsider the sequence {Fn} n≥0 of Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n+2 = F n+1 + Fn for all n ≥ 0. In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 3

    On the multiplicity in Pillai\u27s problem with Fibonacci numbers and powers of a fixed prime

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    Let ( {F_n}_{ngeq 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 (kge 2) and (ellge 0). We prove that (m_{F,p}(c)le 4)

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

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    In this paper, we show that if (X_n, Y_n) is the nth solution of the Pell equation X^2−dY^2= ±1 for some non-square d, then given any int eger c, the equation c = X_n−2^m has at most 2 integer solutions (n, m)withn ≥ 0andm ≥ 0, except for the only pair (c, d)=(−1, 2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences
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