8 problems

For $n\in\mathbb{N}$, define … Let $B_{h_n}(t)$ denote the associated Stern polynomial. No-real-roots conjecture. For each $n\in\mathbb{N}$, the polynomial $B_{h_n}(t)$ is complete…

Let $s_{i,n}$, for $i\in\{0,1,2,3\}$, be the sequences defined earlier in the paper, and let $B_{s_{i,n}}(t)$ be their associated Stern polynomials. Conjectures for $B_{s_{i,n}}$.…

Conjectures for $B_{p_{k,n}}$. (1) If $k\equiv0\pmod 2$, there is $c_k\in\mathbb{N}_{+}$ such that $B_{p_{k,n}}(t)$ has exactly one real root for $n\in\mathbb{N}_{\geq c_k}$; under…

Let … and define … Then the following equality holds: … where $k$ is equal to $n/2$ if $n$ is even and $(n-1)/2$ if $n$ is odd. This conjecture arose from investigations of the seq…

Let $B_n(t)$ denote the Stern polynomial indexed by the positive integer $n$. For a positive integer $n$, say that it has exactly $k$ prime divisors, with the source's convention f…

Let $B_p(t)$ be the Stern polynomial indexed by a prime number $p$. The prime-index irreducibility conjecture. For every prime number $p$, the polynomial $B_p(t)$ is irreducible. T…

The characterization conjecture.

Let $B_n(t)$ denote the Stern polynomial associated with the positive integer $n$. A rational number $a$ is a zero of $B_n(t)$ if $B_n(a)=0$ for some positive integer $n$. The rati…