27 problems

Let ${\bf u}=(u_n)_{n\in\mathbb{N}}$ be a binary sequence that is not eventually periodic. Suppose that the set of return lengths $\mathcal{R}(\mathbf{u})$ is finite, and that … is…

Let $F_m$ denote the $m$th Fibonacci number, and let $\left(\frac{p}{5}\right)$ be the Legendre symbol for a prime number $p$. Wall–Sun–Sun prime conjecture. There are no prime num…

Consider the Fibonacci recurrence $a_n=a_{n-1}+a_{n-2}$ and its chiral recurrence $a_n=-a_{n-1}+a_{n-2}$ with random integer initialization $(a_0,a_1)\in\mathbb Z^2$. Let $F_n$ and…

Let $F_m$ be the set of periodic sequences from the Fibonacci recurrence or its parity transform with initial condition $(a_0,a_1)\in(\mathbb Z/m\mathbb Z)^2$, taken modulo $m$. Fo…

For the Fibonacci recurrence and its parity transform modulo prime powers, Self-Similarity at Prime Powers. At each transition $p^k\to p^{k+1}$, all existing periods are preserved,…

Let $p$ be prime, and let $\#(p)$ count distinct periods modulo $p$ of the Fibonacci recurrence and its parity transform over all initial conditions…

Let $\alpha=(1+\sqrt{5})/2$ be the golden ratio, let $\bf ftm$ be the Fibonacci-Thue-Morse sequence, and let $M_{\bf ftm}(n)$ denote its maximum order complexity at length $n$. Jam…

Simple-normality conjecture. The Fibonacci concatenation is simply normal in every base.

Absolute-normality conjecture. The Fibonacci concatenation is absolutely normal.

Maximum conjecture. If coprime bases $\beta_1$ and $\beta_2$ satisfy $\Upsilon(\beta_1)=u$ and $\Upsilon(\beta_2)=v$, then

Normality conjecture for bases with four zeros. For every base $\beta$ with $\omega(\beta)=4$ and every nonnegative integer $y$, the Fibonacci concatenation is normal in base…

Uniform-distribution conjecture. For every base $\beta$ with $\omega(\beta)=4$, there exists a $K$ such that, for all $k\geq K$, the digits in the sequences…

Wall–Sun–Sun prime conjecture. There are infinitely many Wall–Sun–Sun primes.

Let $F_i$ denote the $i$th Fibonacci number. Consecutive-pair conjecture. There are infinitely many pairs of positive integers $(n,n+1)$ such that … This conjecture asks whether th…

Let $F_i$ denote the $i$th Fibonacci number, and for a positive integer $m$ write $\sum_{i=1}^m F_i$ for the sum of the first $m$ Fibonacci numbers. Shift-extension conjecture. For…

For a positive integer $\ell$ and an integer $m$, let … and define $U_\ell(m)$ analogously. Infinitude conjecture for gap fibers. For every $m\in D_\ell$, … The conjecture expresse…

Let $D_\ell$ and $U_\ell$ be the down-gap and up-gap sets, respectively, defined by … with $U_\ell$ defined analogously. Equality of the Fibonacci gap sets. For every positive inte…

For a positive integer $n$, an $n$-slow Fibonacci walk is a Fibonacci walk $w_k$ attaining the maximum possible number of steps before reaching $n$, denoted by $s(n)$. Let … and le…

Let $F_n$ be the Fibonacci sequence defined by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 2$. Let $\pi(m)$ denote the period of the Fibonacci sequence modulo the positi…

For $h,t\geq2$, consider the admissible residue classes associated with the finite Collatz subsequences $C^h(s)=(T^k(s))_{k=0}^h$ modulo $12\cdot2^h$ and $C^t(s)=(T^k(s))_{k=0}^t$…

Conjecture. These inequalities hold for all $k$ for which they are defined; namely, $M_{k+1}\geq M_k$ and $M_{k+1}\leq M_k+M_{k-1}$ for all $k\geq 2$.

Let $c(n)$ denote the self-Fibonacci divisor function, and let … The authors note that … for $n$ tending to infinity through a set of asymptotic density $1$. Typical-growth conject…

Let $\Phi$ be the set of labeled Fibonacci trees, and let $\vartriangleleft$ be the partial order on $\Phi$ defined by the subtree relation. For $\mathcal{F},\mathcal{F'}\in\Phi$,…

Let $n$ be such that a division-free $n$-free Fibonacci sequence exists, and choose the starting integers from the range $1$ to $N$. A division-free $n$-free Fibonacci sequence is…

Let the starting integers be chosen from the range $1$ to $N$, and consider the resulting 5-free Fibonacci sequence under the paper's random-choice model. A division-free 5-free Fi…