9 problems

Congruences for Domb numbers and Lucas sequences. For every odd prime $p$: (i) if one of the Legendre symbols $(\frac{p}{3})$ and $(\frac{p}{13})$ is $1$, then

Refined moment conjecture. As $x\to\infty$,

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

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…

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…

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…