In this note, we estimate the distance between two $q$-nomial coefficients $\left|{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q-{\left(\genfrac{}{}{0pt}{}{n^{\text{'}}}{k^{\text{'}}}\right)}_q\right|$, where $(n,k)\ne (n^{\text{'}},k^{\text{'}})$ and $q\ge 2$ is an integer.

Let $k\ge 2$ and define $F^{\left(k\right)}:={\left(F_n^{\left(k\right)}\right)}_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{\left(k\right)}=F_{n-1}^{\left(k\right)}+F_{n-2}^{\left(k\right)}+\cdots +F_{n-k}^{\left(k\right)}$, with initial conditions $0,0,\cdots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{\left(k\right)}=1$. The sequences $F:=F^{\left(2\right)}$ and $T:=F^{\left(3\right)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{\left(k\right)}=F_m^{\left(\ell \right)}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{\left(k\right)}\cap F^{\left(m\right)}$ which gives evidence supporting...

We compare the growth of the least common multiple of the numbers $u_{a_1},...,u_{a_n}$ and $|u_{a_1}\cdots u_{a_n}|$, where ${\left(u_n\right)}_{n\ge 0}$ is a Lucas sequence and ${\left(a_n\right)}_{n\ge 0}$ is some sequence of positive integers.

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n+1}=PUₙ-Q{U}_{n-1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n+1}=PVₙ-Q{V}_{n-1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_r\ne 1$. We show that there is no integer x such that $Vₙ=V_{r}Vₘx²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $Vₙ=VₘV_{r}x²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and Q ≡ 1 (mod...

For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by $F_k=2^k$ for $0\le k\le m-1$ and $F_k=F_{k-1}+F_{k-2}+\cdots +F_{k-m}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let $R_m\left(n\right)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_m\left(n\right)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the number of partitions...

The density of primes dividing at least one term of the Lucas sequence ${\left\{L_n\left(P\right)\right\}}_{n=0}^{\infty }$, defined by $L_0\left(P\right)=2,L_1\left(P\right)=P$ and $L_n\left(P\right)=P{L}_{n-1}\left(P\right)+L_{n-2}\left(P\right)$ for $n\ge 2$, with $P$ an arbitrary integer, is determined.

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d=2^{r}d₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ${(b+\surd (b²+4^{\alpha })/2)}^{(p-1)/4)}\left(modp\right)$ for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ${(2a+\surd 4a²+1)}^{(p-1)/4}\left(modp\right)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p-1)/4}\left(modp\right)$ and the criterion for $p|U_{(p-1)/8}$ (if p ≡ 1 (mod 8)), where Uₙ is the Lucas sequence given by U₀ = 0, U₁ = 1 and $U_{n+1}=bUₙ+U_{n-1}\left(n\ge 1\right)$, and b ≢...

For k ≥ 2, the k-generalized Fibonacci sequence $\left(F{ₙ}^{\left(k\right)}\right)ₙ$ is defined to have the initial k terms 0,0,...,0,1 and be such that each term afterwards is the sum of the k preceding terms. We will prove that the number of solutions of the Diophantine equation $F{ₘ}^{\left(k\right)}-F{ₙ}^{\left(\ell \right)}=c>0$ (under some weak assumptions) is bounded by an effectively computable constant depending only on c.