Related papers: A Conjecture concerning the Fibonomial Triangle
We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we…
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…
For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,...,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In 2005, Noe and Post conjectured…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…
We study the $K$-Fibonacci sequence $\mathcal{F}_p$ modulo prime $p$. Cardinalities of sets $|\mathcal{F}_p+\mathcal{F}_p|$ and $|\mathcal{F}_p\cdot\mathcal{F}_p|$ are estimated. We present the method of estimating doubling constant of some…
Much is known about binomial coefficients where primes are concerned, but considerably less is known regarding prime powers and composites. This paper provides two conjectures in these directions, one about counting binomial coefficients…
Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…
We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…
We present two complementary proofs that, if the lengths of $n$ sticks are sampled at random, then the probability that no $p+1$ sticks can form a $(p+1)$-sided polygon can be expressed as the product of the reciprocals of a series of terms…
In 2013, Joerg Arndt recorded that the Fibonacci numbers count integer compositions where the first part is greater than the second, the third part is greater than the fourth, etc. We provide a new combinatorial proof that verifies his…
We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$,…
We study the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single child, also the number 1. We define the tree recursively in the following way: if…
The random Fibonacci chain is a generalisation of the classical Fibonacci substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 01$ with probability $p$ and $1 \mapsto 10$ with probability $1-p$ for $0<p<1$ and where…
The Fibonacci numbers satisfy the famous recurrence $F_n = F_{n - 1} + F_{n - 2}$. The theory of C-finite sequences ensures that the Fibonacci numbers whose indices are divisible by $m$, namely $F_{mn}$, satisfy a similar recurrence for…
If L, respectively R are matrices with entries binom{i-1,j-1}, respectively binom{i-1,n-j}, it is known that L^2 = I (mod 2), respectively R^3 = I (mod 2), where I is the identity matrix of dimension n > 1 (see P10735-May 1999 issue of the…
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…
Zeckendorf proved that every positive integer can be expressed as the sum of non-consecutive Fibonacci numbers. This theorem inspired a beautiful game, the Zeckendorf Game. Two players begin with $n \ 1$'s and take turns applying rules…
The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger…
One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery,…