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In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.
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Let $F_{n}$ be the $n$-th Fibonacci number. Put $φ=\frac{1+\sqrt5}{2}$. We prove that the following inequalities hold for any real $α$: 1) $\inf_{n \in \mathbb N} ||F_nα||\le\frac{φ-1}{φ+2}$, 2) $\liminf_{n\to \infty}||F_nα||\le 1/5$, 3) $\liminf_{n \to \infty}||φ^n α||\le 1/5$. These results are the best possible.
In this paper, we completely solve the Diophantine equations $F_{n_1} + F_{n_2} = 2^{a_1} + 2^{a_2} + 2^{a_3}$ and $ F_{m_1} + F_{m_2} + F_{m_3} =2^{t_1} + 2^{t_2} $, where $F_k$ denotes the $k$-th Fibonacci number. In particular, we prove that $\max \{n_1, n_2, a_1, a_2, a_3 \}\leq 18$ and $\max \{ m_1, m_2, m_3, t_1, t_2 \}\leq 16$.
On Diophantine equations with Fibonacci and Lucas sequences associated to Hardy-Ramanujan problem
该研究围绕哈代-拉马努金问题，探究涉及斐波那契序列和卢卡斯序列的两类三次丢番图方程，找出了兼具哈代-拉马努金数属性的斐波那契数与卢卡斯数。
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, 144 and the only perfect powers in the Lucas sequence are 1, 4.
We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the sequence of $k$-generalized Fibonacci numbers, then there are only finitely many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are all members of $\{F_n^{(k)}: n\ge 1\}$. This generalizes a previous result (cf. arXiv:1508.07760) where the statement for $k=3$ was proved. The result is ineffective since it is based on Schmidt's subspace theorem.
Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers
本研究围绕经典斐波那契恒等式的推广展开，分别在k-广义斐波那契数、普通斐波那契数的幂次及连续项幂和场景下求解三类丢番图方程，推广了已有相关研究结论。
In this paper we focus on finding all the factorials expressible as a product of a fixed number of $2k$-nacci numbers with $k \geq 2$. We derive the 2-adic valuation of the $2k$-nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with $p$-regular sequences.
On a Diophantine Equation with Jacobsthal and Fibonacci Numbers
研究含雅各布斯塔尔数和斐波那契数的丢番图方程，核心方法是对数线性形式技术，主要贡献是确定\(2<k<l<m\)时，丢番图方程\(F_kF_lF_m = J_n\)唯一解为\((k,l,m,n)=(5,7,8,12)\)。
A Pair of Diophantine Equations and Fibonacci-Like Sequences
本文研究两个互素数的丢番图方程解的存在性与唯一性，并针对满足斐波那契递推的序列的连续项，给出具体解及方程选择方法。
Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation
利用韦达跳跃方法，完全分类了有理丢番图方程 (a+1)/b + (b+1)/a = k 的正整数解，证明 k 只能为 3 或 4，且解与斐波那契数相关。
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we characterize.
Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ be 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 of $ a $ and $ b $, we mean the both concatenations $\overline{ab}$ and $\overline{ba}$ together, where $ a $ and $ b $ are any two non negative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations $ F_n=10^d F_m +L_k $ and $ F_n=10^d L_m+F_k $ in non-negative integers $ (n,m,k) ,$ where $ d $ d…
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
A Pair of Diophantine Equations Involving the Fibonacci Numbers
本文针对互素正整数对为斐波那契数平方、立方对的情形，推导了一类丢番图方程唯一非负整数解的显式公式，并构造了存在唯一正整数解的新方程对，得到相关斐波那契数恒等式。
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we have determined all the powers of 2 which are sums of five Fibonacci numbers with few exceptions that we characterize. We have also stated an open problem relating to the number of solutions of equations like those studied in this paper.
Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive integers satisfying $n>m>0$, unless $y=2,3,4,6$ or $10$.
In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use $p$-adic versions of both tools.
In this article, we consider the Diophantine equation $σ_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible in terms of Lucas sequences. Meanwhile, we obtain some results relating to other linear recurrent sequences.
On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers
本文研究了Fibonacci和Lucas数的混合b-拼接形式，通过求解两个丢番图方程确定哪些Lucas数可表示为Fibonacci和Lucas数的拼接，证明仅存在有限多个这样的Lucas数。
In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethö version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.
In this paper, we find all integers $c$ having at least two representations as a difference between linear recurrent sequences. This problem is a pillai problem involving Padovan and Fibonacci sequence. 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.
For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n}$ be the $k-$Fibonacci sequence which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for powers of 2 which are sums of two $k-$Fibonacci numbers. The main tools used in this work are lower bounds for linear forms in logarithms and a version of the Baker--Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of \cite{BL2} and \cite{BL13}.
Fibonacci or Lucas numbers that are products of two Lucas numbers or two Fibonacci numbers
本文研究 Fibonacci 数与 Lucas 数互为两项乘积的丢番图方程，利用 Dujella–Pethö 引理和 Matveev 定理求出全部解及公共项。
Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
In this paper, we study the diophantine equation ${{σ}_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are $n={{F}_{2k+1}}{{F}_{2k+2m+1}}$, where both ${{F}_{2k+1}}$ and ${{F}_{2k+2m+1}}$ are Fibonacci primes. Meanwhile, we show that the twin primes conjecture holds if and only if the equation ${{σ}_{2}}(n)-{{n}^{2}}=2n+5$ has infinitely many solutions.
We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.
We explicitly solve the diophantine equations of the form $$ A_{n_1}A_{n_2}\cdots A_{n_k}\pm 1 = B_m^2 $$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques (2011) and L. Szalay (2012) concerning a variant of Brocard-Ramanujan equation. This is a manuscript of the article published in Communications of the Korean Mathematical Society, 32(3) (2017), pp. 511-522
On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$
本文研究丢番图方程 $F_n = F_l^k (F_l^m-1)$，利用 Matveev 定理、改进的 Baker-Davenport 约化方法及斐波那契数整除性质，证明唯一正整数解为 $(6,3,3,1)$。
Product of powers of distinct primes as sums of Fibonacci numbers
本文研究了不同素数的幂的乘积作为斐波那契数和的Diophantine方程，并确定了满足条件的素数对。
Narayana numbers that are products of two Fibonacci numbers
本文求解丢番图方程 $N_m=F_nF_k$，确定所有可表示为两个斐波那契数乘积的非零 Narayana 数。
We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general method for the solution of exponential diophantine equations. Using a generalization of a result of Erdős, Pomerance and Schmutz concerning Carmichael's $λ$ function, we can make our search systematic for certain moduli needed in the method.
In this paper, we find all solutions of the exponential Diophantine equation $B_{n+1}^x-B_n^x=B_m$ in positive integer variables $(m, n, x)$, where $B_k$ is the $k$-th term of the Balancing sequence.
Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.
On certain $D(9)$ and $D(64)$ Diophantine triples
研究关于$D(9)$和$D(64)$丢番图三元组，推广刻画特定$D(l^2)$-丢番图三元组对应佩尔方程解的结果，利用该结果及代数数对数线性形式的界，对特定形式的$D(9)$和$D(64)$-丢番图三元组进行分类。
In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in Diophantine approximation, due to Dujella and Pethö. This paper extends the previous work of \cite{Patel}.
We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.
Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq 10^{62}$, then the equation has at most two positive integer solutions $(x,y,z)$.
Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\{1,2\}$. Based on the specific cases we could solve, and a computer search with $p,q,k\le100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation.
On Pell numbers representable as product of two generalized Fibonacci numbers
本文研究佩尔数何时可表示为两个 k-斐波那契数之积，利用对数线性形式下界、连分数性质及丢番图逼近工具完全解决该问题，推广了多个已有结果。
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.
We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number problem.
We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $ω_2(ξ)$, $\widehatω_2(ξ)$, $λ_2(ξ)$, $\widehatλ_2(ξ)$ associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new info…
Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z} has only the positive integer solutions (x,y,z)=(a,2,2). In this study, we prove that Terai's conjecture is true when b is a product of two primes and c=5(mod8).
Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$ has at most three positive integer solutions $(x,y,z)$.
On the exponential Diophantine equation $x^2+p^mq^n=2y^p$
本研究结合Lehmer序列本原因子与初等数论方法，探讨含奇素数$p,q$、正整数$x,y,m,n$的指数丢番图方程$x^2+p^mq^n=2y^p$的可解性。
We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.
An Exponential Diophantine equation $x^2+3^α 113^β=y^{\mathfrak{n}}$
本文针对满足互素条件的指数丢番图方程 $x^2+3^α113^β=y^{\frak{n}}$，开展其在正整数变量、非负指数及不小于3的整数指数下的全部解的确定研究。
Families of Unit Equations and Exponential Diophantine Problems via Integral Points
本文通过Ru-Vojta定理和Huang-Levin-Xiao不等式推广研究了投影簇上整数点的分布，证明了两类一参数单位方程解集的退化性，并扩展了GCD估计以获得特定指数迪菲antine方程和q进制表示中数字分布的新结果。
Let $r\ge 1$ be an integer and ${\bf U}:=(U_{n})_{n\ge 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\ge 0 $. In this paper, we show that there are no positive integers $r\ge 3,~x\ne 2,~n\ge 1$ such that $U_n^x+U_{n+1}^x$ is a member of ${\bf U}$.
Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture
本文研究具有四个连续底数项的纯指数丢番图方程，证明其仅存在两组正整数解，并对$n\ge4$的情况给出显式模数证明无解，为斯科伦关于纯指数丢番图方程局部-全局原理的经典问题提供了贡献。
Let $\left\lbrace F_{k}\right\rbrace_{k\geq0}$ be the Fibonacci sequence defined by $F_{k}=F_{F-1}+F_{k-2}$ for all $ n\geq2$ with initials $F_{0}=0\; F_{1}=1$. Let $\left\lbrace J_{n}\right\rbrace_{n\geq0}$ be the Jacobsthal sequence defined by $J_n=2J_{n-1}+J_{n-2}$ for all $ n\geq2$ with initials $J_0=0$, $J_1=1$. In this paper we find all the solutions of the two Diophantine equations $J_n +J_m =F_a$ ,$F_n +F_m =J_a$ in the non-negative integer variables (n,m,a),i.e we determine all Fibonacci numbers which are sum of two Jacobsthal numbers, and also determine all Jacobsthal numbers which a…
In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1<a<b<101 with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation in the title. It is proved that the equation (a^n-1)(b^n-1)=x^2 has no solutions if a,b have opposite parity and n>4 with 2|n. Also, we solved (a^n-1)(b^n-1)=x^2 for the pairs (a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49), and (45,100). Lastly, we show that when b is even, the equation (a^n-1)(b^(2n)a^n-1)=x^2 has no solutions n,x.
In this paper we obtain three undecidable results for exponential diophantine equations over the field $\mathbb Q$ of rational numbers. For example, we prove that there is no algorithm to decide the solvability of a general exponential diophantine equation $F(x_1,\ldots,x_{8})=0$ over $\mathbb Q$ with eight unknowns.
Let $\ell, m, r$ be fixed positive integers such that $2\nmid \ell$, $3\nmid \ell m$, $\ell>r$ and $3\mid r$. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if $\min\{r\ell m^2-1,(\ell-r)\ell m^2+1\}>30$, then the equation $(r\ell m^2-1)^x+((\ell -r)\ell m^2+1)^y=(\ell m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.
In [1] it is shown that the Diophantine equation $(k!)^n+k^n=(n!)^k+n^k$ only has the trivial solution $n=k$, and $(k!)^n-k^n=(n!)^k-n^k$ only has the solutions $n=k$, $(n, k)=(1, 2),$ and $(2, 1)$. In this article we find all solutions of the Diophantine Equations $a_1!a_2!\cdots a_n! \pm a_1a_2 \cdots a_n = b_1!b_2! \cdots b_k! \pm b_1b_2 \cdots b_k$, where $a_i$ majorizes $b_i$. Furthermore we find a sufficient condition on a function $f:N\to R^+$ to guarantee that $f$ gives a monotone function on the POSET of all finite sequences of natural numbers. We then use that to solve other Diophant…
Solving an infinite number of purely exponential Diophantine equations with four terms
本文研究了具有连续基数的四项纯指数丢番图方程的无限族，证明了当 $n \equiv 3 \pmod{4}$ 时方程 $n^x+(n+1)^y+(n+2)^z=(n+3)^w$ 的所有正整数解，结合了初等同余论证、Baker 方法和基于 Skolem 猜想变体的算法。
In this paper, we consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z},$ where $a, b, c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{r}, r\in Z^{+}, 2\mid r$ with $b$ even. That is $$a=\mid Re(m+n\sqrt{-1})^{r}\mid, b=\mid Im(m+n\sqrt{-1})^{r}\mid, c=m^{2}+n^{2},$$ where $m, n$ are positive integers with $m>n, m-n\equiv1(mod 2),$ gcd$(m, n)=1.$ $(x, y, z)= (2, 2, r)$ is called the trivial solution of the equation. In this paper we prove that the equation has no nontrivial solutions in positive integers $x, y, z$ when $$r\equiv 2(mod 4), m\equiv 3(mod 4), m>\max\…
On the Exponential Diophantine Equation $(a^n-1)(b^n-1)=x^2$
本文纠正了先前文献的疏漏，并证明在特定条件下（$b$ 为模 8 余 3 的素数，$a$ 为偶数）指数丢番图方程无正整数解，推广了 Szalay 的结果。
In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are $1,2$, and $3$. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another Pell number are $1$ and $2$. Here we add to these two results and prove that for any fixed natural number $P\geq 3$, if we define the sequence $\left(u_n\right)_n$ as $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, then the only solution to the Diophantine equation $φ\left(u_n\right)=u_m$ is $φ\left(u_1\right)=φ(1)=1=u_1$.
Let $A$, $B$ be fixed positive integers such that $\min\{A,B\} > 1$, $\gcd(A,B) = 1$ and $AB \equiv 0 \bmod 2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier \cite{BHV}, we prove that if $A > 8 B^3$, then the equation $(*) \quad (A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$. Combining the above conclusion with some existing results, we can deduce that if $A >8 B^3$ and $B \equiv 2 \bmod 4$, then (*) has o…
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