23 problems

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 $n$ 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 $(a,b)$ be an $(a,b)$-Fibonacci sequence, and suppose it falls under case (vi) of the finite-orders conjecture, namely the case where the range of $\omega_{(a,b)}(m)$ is conjec…

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…

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…

Let $F$ denote the Fibonacci numbers, let $AP_F$ denote the associated arithmetic-progression sequence, and let $\Delta(L,k;r)$ and $n(AP_F,k;r)$ be the parameters defined in the p…

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 $a_0=1$, $a_1=2$, and define $a_n=3a_{n-1}-a_{n-2}$. For ${\boldsymbol{i}}=(i_1,i_2)\in\langle a_n\rangle\times\langle a_{n+1}\rangle$, let $u(\alpha({\boldsymbol{i}}))$ be the…

Soydan–Németh–Szalay conjecture. Apart from these trivial solutions, the only solutions are

Fibonomial Lucas-type divisibility conjecture. If $p^*\geq p$, then

Let an $(s,t)$-core partition be a partition that is simultaneously $s$-core and $t$-core, and let $F_n$ denote the $n$th Fibonacci number. Amdeberhan's conjecture. The number of…

Canonical-form conjecture. For all $n\in\mathbb{N}$,

Melham's conjecture. For any positive integers $n,m$, there is a polynomial $P_{2m-1}(x)$ of degree $2m-1$ with integer coefficients such that

Bruckman–Anderson conjecture. If $m=q^e$ is a prime power with $e\geq 1$, then

Let $F_n$ denote the $n$th Fibonacci number, let $up$ be the $p$-adic valuation, and let $\alpha(p)$ be the least positive integer $n$ such that $p$ divides $Fn$. For a sequence, i…

Let $F_n$ and $L_n$ denote the Fibonacci and Lucas sequences, respectively. A sequence term is positive when its value is positive, and a term is odd when it is not divisible by…

Let $F_n$ denote the Fibonacci sequence. A Fibonacci number is positive when its value is positive, and an odd Fibonacci number is one not divisible by $2$. Sun's prime–Fibonacci-p…

Let $F_n$ denote the Fibonacci numbers, and let … be the $n$th Catalan number. A Fibonacci number is positive when it is positive as an integer. Hou and Zeng's prime–Fibonacci–Cata…

Let $F_0=0$, $F_1=1$, and $F_{i+1}=F_i+F_{i-1}$ for $i=1,2,3,\ldots$ be the Fibonacci sequence. A Fibonacci number is positive when its index is positive and its value is positive.…

Fibonacci recurrence-coefficient conjecture. If $\mu_n=NCP_2(n)$, then $b_0=b_1=\lambda_1=1$, and for $n\geq2$,