On $k$-generalized Fibonacci numbers which are perfect powers of Lucas numbers

Let (Fn)n≥0and (Ln)n≥0be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations ofaandb, we mean the both concatenationsabandbatogether, whereaandbare any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equationsFn= 10dFm+LkandFn= 10dLm+Fkin nonnegative integers (n,m,k), whereddenotes the number of digits ofLkandFk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.

Diophantische Approximationen

Let $(L_n)_{n\geq 0}$ be the Lucas sequence given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1}+L_n$ for $n \geq 0$. In this paper, we are interested in finding all powers of three which are sums of two Lucas numbers, i.e., we study the exponential Diophantine equation $L_n + L_m = 3^{a}$ in nonnegative integers $n, m,$ and $a$. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.

In this paper, we find that all Mulatu numbers, which are concatenations of two Lucas numbers are 11,17,73,118. Let 〖(M_k)〗_(k≥0) and 〖(L_k)〗_(k≥0) be the Mulatu and Lucas sequences. That is, we solve the Diophantine equation M_k=L_m L_n=10^d L_m+L_n in non-negative integers (k,m,n,d), where d denotes the number of digits of L_n. Solutions of this equation are denoted by (k,m,n,d)=(4,1,1,1),(5,1,4,1),(8,4,2,1),(9,1,6,2). In other words, we have the solutions M_4=L_1 L_1=11, M_5=L_1 L_4=17, M_8=L_4 L_2=73, M_9=L_1 L_6=118. The proof based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of this Diophantine equation.

In this paper, we give all solutions of the Diophantine equation [Formula: see text], where [Formula: see text] x [Formula: see text] x [Formula: see text], [Formula: see text] is the Perrin sequence, and [Formula: see text] is the Tribonacci sequence. We show that this Diophantine equation has only [Formula: see text] integer solution triples. For the proof, we use Baker’s method. Our motivation is to show that linear forms in logarithms can still be effectively used for the solutions of different Diophantine equations involving classical number sequences such as Fibonacci or Lucas sequences.

Let {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ ℤ) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form uk = xn+c which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over ℚ. We illustrate this approach by showing that the quaternary Diophantine equation x2n±6xn + 1 = 8y2 has no solutions in positive integers x, y, n with x, n > 1.

Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in p-adic logarithms. We then use these ideas to improve known estimates on solutions of S-unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety A2(2).

Linear forms in elliptic logarithms

Let (Ln)n ≥ 0 be the Lucas sequence defined by Ln+2 = Ln+1 + Ln for all n ≥ 0, with initial values L0 = 2 and L1 = 1. In this paper, we find all the Fibonacci numbers 2Fn and sums of two Fibonacci numbers which are close to a power of 3. As a corollary, we determine all Lucas numbers close to a power of 3. To prove our results, we will use Baker's theory lower bound for linear forms in logarithms of algebraic numbers, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

Let $(F_{r}^{(k)})_{r\geq2-k}$ and $(L_{r}^{(k)})_{r\geq2-k}$ be generalizations of the Fibonacci and Lucas sequences, where $k\geq2$. For these sequences the initial $k$ terms are $0, 0, \ldots , 0, 1$ and $0, 0, \ldots , 2, 1$, and each subsequent term is the sum of the preceding $k$ terms. In this paper, we determined all first and second kinds of Thabit numbers that can be expressed as the sums of $k$-Fibonacci and $k$-Lucas numbers. We employed the theory of linear forms in logarithms of algebraic numbers and a reduction method based on the continued fraction.

In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in diophantine approximation.

Let $$\{F_{r}\}_{r\geq0}$$, $$\{L_{r}\}_{r\geq0}$$ and $$\{Le_{r}\}_{r\geq0}$$ be $$r$$-th terms of Fibonacci, Lucas and Leonardo sequences, respectively. In this paper, we determined the effective bounds for the solutions of the Diophantine equation $$F_{r}=\eta^{k}Le_{s}+L_{t}$$ in non-negative integers $$r$$, $$s$$, $$t$$, where $$k$$ represents the number of digits of $$L_{t}$$ in base $$\eta\geq2$$. In addition, we applied linear forms in logarithms of algebraic numbers and the reduction method based on the continued fraction. In particular, we investigated all solutions of this Diophantine equation for $$\eta\in[2,10]$$.

This expository paper explores the theory of linear forms in logarithms and its applications to Diophantine equations. We begin with foundational results on transcendental numbers, including Liouville's theorem and the Gelfond-Schneider theorem, before developing Baker's theory of linear forms in logarithms. The paper concludes with applications to Diophantine equations through the Baker-Davenport method, illustrating these techniques with concrete examples. The purpose of this paper is to provide a step-by-step understanding of this particular area of Mathematics, and was written as a part of Euler Circle's Independent Paper and Research Writing Program, in which the author studied the topic independently and wrote this paper over a period of 4 weeks.