Mahadi Ddamulira, Florian Luca
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 ${\bf U}$.
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In this article the Diophantine equations of the form $x^{2}-8C_{n}xy+ 16y^{2}=\pm2^{r}$, $x^{4}-8C_{n}x^{2}y+ 16y^{2}=\pm2^{r}$ and $x^{2}-8C_{n}xy^{2}+ 16y^{4}=\pm2^{r}$ are taken into consideration for the investigations of existence of any positive integer solutions. Furthermore, the solutions are manifested in terms of the balancing, Pell and Lucas-Pell numbers.
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.
On a Diophantine Equation Involving Lucas Numbers
本文研究了涉及卢卡斯数的丢番图方程，证明了在特定条件下该方程无解，采用了因式分解、数论引理和计算机辅助计算等方法。
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 $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$ are given integers with $A\ne 0,~B\ne 0$, $m,n,m_1,n_1$ are nonnegative integer unknowns and $r$ is also unknown.
Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we explore the presence of repdigits in the product of consecutive balancing or Lucas-balancing numbers.
We show that the product of two consecutive Fibonacci (respectively Lucas) numbers is divisible by the sum of their indices if this sum is a prime number different from 5 and in the form (4r+1)(respectively (4r+3)).
On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers
本文完整求解了由三个连续$k$--广义卢卡斯数幂构成的非线性丢番图方程，给出了所有非负整数解。
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive te…
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。
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 the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems, including Ingham's exponential theorem as well as some new results. Diophantine inequalities and their connection with Cramer's and Andrica's conjectures are also discussed.
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.
In this paper, we prove a theorem about the integer solutions to the Diophantine equation $x^{4}-q^{4}=py^{r}$, extending previous work of K.Gy\H ory, and F.Luca and A.Togbe, and of the author.
We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.
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.
Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.
We prove that the Diophantine equation N X^2 + 2^L 3^M = Y^N has no solutions (N,X,Y,L,M) in positive integers with N > 1 and gcd(NX,Y) = 1, generalizing results of Luca, Wang and Wang, and Luca and Soydan. Our proofs use results of Bilu, Hanrot, and Voutier on defective Lehmer pairs.
We solve the two Diophantine equations $P_k=J_n+J_m$ and $Q_k=J_n+J_m$ where $\left\lbrace P_{k}\right\rbrace_{k\geq0}$, $\left\lbrace Q_{k}\right\rbrace_{k\geq0}$ and $\left\lbrace J_{k}\right\rbrace_{k\geq0}$ are the sequences of Pell numbers, Pell-Lucas numbers and Jacobsthal numbers, respectively. The main tool is the theory of linear forms in logarithms.
On a problem of Pillai involving S-units and Lucas numbers
本文研究卢卡斯数与S-单位之差构成的指数丢番图方程，分别对c为正整数、负整数和零的情形分类讨论，并完整确定了使方程至少有三个解的所有整数c。
The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.
On the Diophantine Equations $J_n +J_m =L_k$ and $L_n +L_m =J_k$
本文利用 Baker 对数线性形式定理及 Dujella 和 Pethő 的缩减方法，求出了丢番图方程 $L_n +L_m =J_k$ 和 $J_n +J_m =L_k$ 的所有非负整数解，找出了所有可表示为两个 Jacobsthal 数之和的 Lucas 数及反之。
On the Diophantine Equations Arising from Three-Term $b$-Concatenations in Pell-Type Sequences
本文研究Pell型序列中三项$b$进制拼接的丢番图方程，在中间块索引限制下给出上界，并确定$2\leq b\leq15$时的全部解数。
Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$ divides $F_k$. Recently, D. Marques has obtained a formula for $z(F_{n}F_{n+1})$, $z(F_{n}F_{n+1}F_{n+2})$, and $z(F_{n}F_{n+1}F_{n+2}F_{n+3})$. In this paper, we extend Marques' result to the case $z(F_{n}F_{n+1}\cdots F_{n+k})$ for every $4\leq k \leq 6$. We also give a formula for $z(L_nL_{n+1}\cdots L_{n+k})$ when $k = 5,6$ which extends the recent result of…
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。
For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers that appear in different generalized Lucas sequences; i.e., we study the Diophantine equation $L_n^{(k)}=L_m^{(\ell)}$ in nonnegative integers $n,k,m,\ell$ with $k, \ell\geq 2$. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport reduction method. This paper is a continuation…
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.
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 $φ\left(u_1\right)=φ(1)=1=u_1$.
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。
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
In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are different fixed positive integers. We solve the above equation when $m=1$ and $(a,b)=(2,10),(4,100),(10,58),(3,45)$. Moreover, we show that $(a^2-2)(b^2-2)=x^2$ has no solution n,x if 2|n and gcd$(a,b)=1$. We also give a conjecture which says that the equation $(2^2-2)((2P_k)^n-2)=x^2$ has only the solution $(n,x)=(2,Q_k)$, where $k>3$ is odd and $P_k,Q_k$ …
By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map is defined and we show that its three-dimensional representation is also a helix. After proving that the lattice points of these later helix satisfy certain diophantine Pell's equations we call it a Pell's helix. We prove that for P-Fibonacci and Pell's helices the respective ratio is an invariant, contrasting to the P-Lucas helices whose ratio depends on P. …
On the Diophantine Equation $p^x+ (2p+1)^y =z^2$ with Consecutive Exponents
该研究针对所有奇素数 $p$ 的丢番图方程 $p^x+ (2p+1)^y =z^2$，证明其唯一解为 $(2,1,p+1)$（$2p+1$ 非合数时），并对 $2p+1$ 为合数的情况归约为有限多情形。
We study Gibonacci sequences mod $m$, giving special attention to the Lucas numbers. It is known which $m$ have the property that the Fibonacci sequence contains all residues mod $m$. When $m$ has this property, we say that the Fibonacci sequence is complete mod $m$. We extend this work to all Gibonacci sequences, concluding by determining the set of $m$ in which a generic Gibonacci sequence containing the relatively prime consecutive terms $a, b$ is complete mod $m$.
On Pillai's Problem involving Lucas sequences of the second kind
本文研究第二类Lucas序列与完全幂之差的Pillai型丢番图方程，证明在至少存在三个解时，解的大小和特征多项式系数均有上界。
Tribonacci Numbers That Are Products of Two Lucas Numbers
本文研究了Tribonacci数作为两个Lucas数乘积的条件，并证明了相关结论。
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…
We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…
Diophantine equations involving powers of factorials
本文受Alzer和Luca关于阶乘幂丢番图方程整数解工作的启发，研究了两类推广的阶乘幂方程，并证明了类似的整数解结论。
We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} φ(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $φ$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c \rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.
In this paper, we consider the equations involving Euler's totient function $ϕ$ and Lucas type sequences. In particular, we prove that the equation $ϕ(x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a+1, a, 1, 1)$, where $a$ is a positive integer, and the equation $ϕ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a>b\ge 1$.
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.
Diophantine tuples and product sets in shifted powers
本文研究了具有D_k(n)性质的丢番图元组，改进了多个已有结果，并探讨了其在非平凡移位幂集中的应用，结合筛法、丢番图逼近和极值图论思想。
The Diophantine equation $a \left(\frac{b^k-1}{b-1}\right) = \mathcal{U}_n-\mathcal{U}_m$
该研究刻画了一个含广义 Lucas 序列项之差与几何和的丢番图方程，并在判别式为正的整数参数条件下给出其全部正整数解。
Let k>=2 and let (Q_{n}^{(k)})_{n>=2-k} be the k-generalized Pell sequence defined by Q_{n}^{(k)}=2Q_{n-1}^{(k)}+Q_{n-2}^{(k)}+...+Q_{n-k}^{(k)} for n>=2 with initial conditions Q_{-(k-2)}^{(k)}=Q_{-(k-3)}^{(k)}=...=Q_{-1}^{(k)}=0, Q_{0}^{(k)}=2,Q_{1}^{(k)}=2. In this paper, we solve the Diophantine equation Q_{n}^{(k)}=y^{m} in positive integers n,m,y,k with m,y,k>=2. We show that all solutions (n,m,y) of this equation in positive integers n,m,y,k such that 2<=y<=100 are given by (n,m,y)=(3,2,4),(3,4,2) for k>=3. Namely, Q_{3}^{(k)}=16=2^4=4^2 for k>=3.
Creating Decidable Diophantine Equations
本文推广Matiyasevich的论证，提出一种生成无穷多个解可由线性递推完全刻画的丢番图方程的方法，并构造出唯一整数根为Tribonacci数连续三元组的整系数多项式。
Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most $50$ consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most $50$ consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.
On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers
该研究针对Pell序列与Pell-Lucas序列相关的两类混合连乘丢番图方程，证明其非负整数解仅有有限多组，并明确给出了进制数2到10范围内的所有解。
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…
On generalized Thabit numbers $(p+1)p^\mathfrak{a}-1$ in the $k$-Lucas sequence
本文研究了$k$-Lucas序列中广义Thabit数$(p+1)p^\mathfrak{a}-1$的解，针对Mersenne或Fermat素数$p=2^{\ell}\pm 1$及正整数$n,k,\mathfrak{a},\ell$的解法。
In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where $m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and $\gcd(x,y)=1.$ We prove that there are only finitely many solutions $(m,p,q,x,y)$ for which $y$ is not a sum of two consecutive squares. We also study the above equation with fixed $y$ and with fixed $q.$
Some Diophantine equations involving arithmetic functions and Bhargava factorials
本文将 Luca 关于阶乘与 Euler 函数、除数和及除数计数函数的丢番图方程有限解结论，推广到多阶乘乘积和除数 k 次幂和函数，并扩展至 Bhargava 阶乘。
Diophantine equations involving double factorials
受Alzer与Luca关于阶乘丢番图方程解的结果启发，本文研究双阶乘形式的同类丢番图方程及其变体的所有整数解，并证明在ABC猜想下一类含双阶乘的有理多项式方程仅有有限个整数解。
In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + \cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 \leq d \leq 50$.
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。
We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers. We prove that such an intersection is finite except for the case $U_n(1,-1)$ and $U_n(3,1)$ and the case of two $V$-sequences when the product of their discriminants is a perfect square. Moreover, the intersection in these cases also forms a Lucas sequence. Our approach relies on solving homogeneous quadratic Diophantine equations and Thue equation…
In a simple integer chain, if $u_{i-1}$, $u_i$, and $u_{i+1}$ are three consecutive terms of the chain, and the pair $(u_{i-1}, u_i)$ has a certain property, then the next pair $(u_i, u_{i+1})$ also has the same property. We extend the idea of a simple chain to an $n$-chain in which $n$ is a positive integer and if a pair $(u_{i-1}, u_{i})$ has a certain property, then the $n$th next pair $(u_{\pm n+i-1}, u_{\pm n+i})$ also has the same property. In this case, we call $(u_{i-1}, u_{i}, u_{i+1})$ and $(u_{\pm n+i-1}, u_{\pm n+i}, u_{\pm n+i+1})$ matching triples. We use $4$-chains to study a fa…
Square roots $s$ of sums of $M$ consecutive integer squares starting from $a^{2}\geq1$ are integers if $M\equiv0,9,24$ or $33(mod\,72)$; or $M\equiv1,2$ or $16(mod\,24)$; or $M\equiv11(mod\,12)$ and cannot be integers if $M\equiv3,5,6,7,8$ or $10(mod\,12)$. Finding all solutions with $s$ integer requires to solve a Diophantine quadratic equation in variables $a$ and $s$ with $M$ as a parameter. If $M$ is not a square integer, the Diophantine quadratic equation in variables $a$ and $s$ is transformed into a generalized Pell equation whose form depends on the $M(mod\,4)$ congruent value, and who…
On Wagstaff primes in the $k$-Lucas number sequence
本文研究了k-卢卡斯数列中瓦格斯素数的分布，证明了特定条件下的唯一解。
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.
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