60 problems

Let $\mathcal{F}$ denote the concatenated Fibonacci constant in base $10$. A real number is normal in base $10$ if every block of $k$ digits occurs with limiting frequency…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence, defined by $F_0=0$, $F_1=1$, and … A perfect power is a positive integer of the form $y^a$ with integers $y>1$ and $a>1$. Luca–Pate…

Let $m$ be a positive integer. The Pisano period modulo $m$ is one shortest period of the Fibonacci sequence modulo $m$, and its number of zeros is the order of $m$. OEIS conjectur…

Fibonacci squared-sum gcd conjecture. For all even $k$,

Let $\mathbb P$ be a piece, and let a twisted Fibonacci spiral be the vertex configuration defined by four selected moves of $\mathbb P$ in the Fibonacci-spiral pattern, together w…

Let $F_q$ be the $q$th Fibonacci number and let $[F_q]=\{1,2,\ldots,F_q\}$. For the inside-out polytope associated with $q$ queens, let $D$ denote its denominator. Queen denominato…

Let $F_q$ be the $q$th Fibonacci number, let $[F_q]=\{1,2,\ldots,F_q\}$, and let $u_{\mathbb Q}(q;n)$ be the counting quasipolynomial for $q$ queens. Kotěšovec's conjecture. The pe…

Let $F_n$ be the $n$th Fibonacci number, let $e(n,i,k)$ be the exponent used to determine the sign in the source, and let $d_k(r)$ be the Fibonomial coefficient corresponding to…

Let $F_n$ denote the $n$th Fibonacci number. A Fibonomial coefficient is obtained by replacing the binomial coefficients in the corresponding reciprocal Hankel matrices with Fibono…

Let $f(n)$ be the number of representations of a positive integer $n$ as a sum of distinct Fibonacci numbers. For a representation $n=\sum_{j=1}^k\epsilon_jF_j$ with…

Let $F_n$ denote the $n$th Fibonacci number, defined by … where $\alpha$ and $\beta$ are the roots of $t^2-t-1$. For positive integers $n,k,p,q$, consider … The solutions…

The converse conjecture. If $S(q)\neq\emptyset$, then there exists a prime $p\equiv1\pmod q$ such that $z(p)\mid\pi(q)$.

Let $F=\{F_n:n\geq 1\}$ be the set of Fibonacci numbers, where $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$. For a set $A\subseteq\mathbb{Z}$, let $dl^\infty(A)$ denote its…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence, defined by $F_0=0$, $F_1=1$, and … A Fibonacci prime is a Fibonacci number $F_n$ that is prime. The Fibonacci-prime infinitude conj…

Consider the Ordered Zeckendorf game starting from $n$ copies of $F_1$, with each legal move selected uniformly at random. Random-game distribution conjecture. As $n\to\infty$, the…

Let $n$ be the initial number of copies of $F_1$ in the Ordered Zeckendorf game, and let the Long Game Strategy (LGS) be the priority-based strategy that performs switch moves, com…

Let $a,b,c,\ell,r,q,u$ and $\mu'$ be as defined above, and let $\mathcal{X}_{dist}$ be the set of distances between consecutive elements of the ordered set $\mathcal{X}_{ord}$. Sup…

Let $\mathcal{P}$ be the set of positive primes, let $\mathbb{N}$ be the set of positive integers, and let $\{f_n\}$ be the Fibonacci sequence defined by $f_0=0$, $f_1=1$, and…

Fibonacci perfect-power conjecture. The only Fibonacci numbers that are perfect powers are $0$, $1$, $8$, and $144$. This conjecture was confirmed by Bugeaud, Mignotte and Siksek u…

Complete-solution conjecture. The only solutions are

Erduvan–Keskin conjecture. This equation has no solutions when $y$ is a prime number $>7$. This conjecture extends the known results for the fixed bases $y=2,3,5$ and leaves the ge…

Let $(a,b)$-Fibonacci sequences be defined by $\mathcal{U}_0=0$, $\mathcal{U}_1=1$, and $\mathcal{U}_n=a\mathcal{U}_{n-1}+b\mathcal{U}_{n-2}$. For $m>1$, let $\omega_{(a,b)}(m)$ be…

Let $(a,b)$-Fibonacci sequences be defined by $\mathcal{U}_0=0$, $\mathcal{U}_1=1$, and $\mathcal{U}_n=a\mathcal{U}_{n-1}+b\mathcal{U}_{n-2}$. Let $\omega_{(a,-1)}(m)$ be the numbe…

Let $K$ be even, and let $F_{K,n}$ be the $K$-Fibonacci sequence. For a modulus $m$, let $\pi_K(m)$ denote its Pisano period, let $\omega_K(m)$ denote the number of zeros in that p…

The super FiboCatalan numbers are defined by replacing the binomial and Catalan-number expressions in the corresponding super Catalan identities with their FiboCatalan analogues. A…