16 problems

Chebyshev-like bias conjecture. (Weak) If $U$ is a regular Lucas sequence with $D>0$, then

Let $(V_n(p,q))_{n\geq 0}$ be the Lucas sequence considered in the paper, and let $\pi_V(m)$ denote its period modulo $m$. The vertical slice result refers to the additive-inverse…

Let $U_n:=U_n(p,q)$, let $m\geq 1$, and let $n\in\mathbb{Z}$. Assume that both $p$ and $m$ are even. If … then exactly one of the following conclusions occurs:…

Gordon–Pomerance conjecture. With the same hypotheses as Lemma Gordon–Pomerance, the displayed upper bound holds. The conjecture is used to obtain the conditional upper bound for t…

Let $s\in\mathbb Z$. The sequence $(P_n(s,0))_{n\geq 1}$ is called polynomially recurrent if it satisfies a polynomial recurrence in $n$. Polynomial-recurrence conjecture. The sequ…

Let $(u_n)_{n\geqslant 0}$ be a Lucas sequence, and let $g_u(n)$ denote the gcd function associated with it. Assume that $\Delta_u\ne 1$. Then the maximal gcd conjecture asserts th…

Let $(u_n)_{n\ge 0}$ be a Lucas sequence of integers satisfying … for all $n\ge 0$, with $u_0=0$, $u_1=1$, where $r$ and $s$ are nonzero coprime integers, $r^2+4s\ne 0$, and the qu…

For integers $P,Q$, define the Lucas sequence $(U_n(P,Q))$ by … The sequence is said to satisfy the Dold condition when its associated Dold congruences hold. Lucas-sequence conject…

Complete-solution conjecture. The only solutions $(r,s,n)$ with $n\geq4$ are

Complete-solution conjecture. The solutions of this equation are exactly the listed parameter tuples: $(n;r,s)=(12;\pm1,1)$ with $U_n=\pm144$; $(n;r,s)$ in

Finiteness conjecture. There are only finitely many solutions of this equation with $n\in\{6,12\}$, whether $\alpha,\beta$ are real or complex conjugates.

Conjecture 2. The values $U_{(p-1)/4}(b,-1)$ and $V_{(p-1)/4}(b,-1)$ satisfy

Conjecture 1. If $4\nmid xy$, then $V_{(p-1)/4}(b,-1)\equiv0\pmod p$ and

Finiteness conjecture. There are only finitely many solutions of this equation with $n\in\{3,6\}$. This is presented as an unresolved finiteness assertion for the specified indices…

Pairwise coprimality conjecture. For any two different primes $p$ and $q$, the terms $u_p$ and $u_q$ are relatively prime:

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…