Related papers: Perfect Numbers and Fibonacci Primes (II)

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.…

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we…

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions…

For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine…

We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation \[ n^x=(n+1)^y+(n+2)^z+(n+3)^w \] in positive integers $n,x,y,z$ and $w$ are given by…

Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove…

We study the number of solutions $N(B,F)$ of the diophantine equation $n_1n_2=n_3n_4$, where $1\le n_1\le B$, $1\le n_3\le B$, $n_2, n_4\in F$ and $F\subset [1,B]$ is a factor closed set. We study more particularly the case when $F=…

For $2k \leq n+1$, we exhibit non-polynomial solutions to the Hessian equation \[ \sigma_{k}(D^{2}u)=1 \] on all of $\mathbb{R}^{n}.$

Let $k,l\geq2$ be fixed integers. In this paper, firstly, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(lx)^{k}=y^{n}$ in integers $x,y,n$ with $x,y\geq1, n\geq2$ satisfy $n<C_{1}$ where $C_{1}=C_{1}(l,k)$ is an…

Let $(a, b, c)$ be a primitive Pythagorean triple parameterized as $a=u^2-v^2,\ b=2uv,\ c=u^2+v^2$,\ where $u>v>0$ are co-prime and not of the same parity. In 1956, L. Je{\'s}manowicz conjectured that for any positive integer $n$, the…

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

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,…

Let $k\ge 2$ and $\{F_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$--generalized Fibonacci numbers whose first $k$ terms are $0,\ldots,0,0,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we determine all…

In this paper we consider the Diophantine equation $ V_n - b^m = c $ for given integers $ b,c $ with $ b \geq 2 $, whereas $ V_n $ varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the…

We study the problem of finding positive integers $n$ such that all the decimal digits of $2^n$ are even, i.e., belong to $\{0, 2, 4, 6, 8\}$. Computational checks up to $n = 10^{15}$ reveal the known cases $n = 1, 2, 3, 6, 11$ and no…

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 $f_1=1,f_2=2$ and $f_i=f_{i-1}+f_{i-2}$ for $i>2$ be the sequence of Fibonacci numbers. Let $\Phi_h(n)$ be the quantity of partitions of natural number $n$ into $h$ different Fibonacci numbers. In terms of Zeckendorf partition of $n$ I…

Solutions of a diophantine equation $f(a,b) = g(c,d)$, with $a,b,c,d$ in some finite range, can be efficiently enumerated by sorting the values of $f$ and $g$ in ascending order and searching for collisions. This article considers functions…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer…

We study the Diophantine equation $\displaystyle{\tfrac{a+1}{b} + \tfrac{b+1}{a} \ = \ k}$, where $k$ is an integer. Using Vieta jumping, we completely classify all positive integer pairs $(a, \, b)$. We prove that the associated integer…