22 problems

Möbius–Chebyshev conjecture. For all integers $i\leq j$, $\mu(a^i,c^j)$ is the coefficient of $x^{j-i}$ in $T_{i+j}(x)$.

Let $T_m(x)$ denote the Chebyshev polynomial of index $m$, and let $\ell$ be a non-zero multiple of $4$. Consider … A continued fraction is semi-specializable if it has the partial…

For a prime $p$, let $R_p$ denote the relevant Chebyshev residue structure, and let $\omega_a$ be the element associated with $a\in R_p$. A Chebyshev primitive root is an element…

Let $f\in \mathbb{C}[x]$ be a complex polynomial, and let $G_\infty(\mathbb{C},f,t)$ denote its geometric iterated Galois group over a transcendental parameter $t$. A polynomial is…

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq1$, let $D_n^{\mathit{Mb}}(d,w,x,y,z)$ be the Gram determinant of type $\mathit{Mb}$. Let $T_k$ and $S_k$ denote the Chebyshev polyn…

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq 1$, let $D_n^{(Mb)_1}$ denote the Gram determinant of type $(Mb)_1$, where the Gram matrix is formed from crossingless-connection d…

Let $Mb_{n,1}$ be the set of crossingless-connection diagrams between $2n$ marked boundary points of the Möbius band, with exactly one curve intersecting the crosscap. Let…

Let $0<\alpha<1$ and let $0\leq\beta\leq 1-\alpha$. Write $\tau_{\alpha,\beta}$ for the asymptotic degree ratio associated with approximating a Chebyshev polynomial after the affin…

Merino's irreducibility conjecture. This polynomial is irreducible over $\mathbb{R}$ if and only if $m$ and $n$ are coprime. The claim gives a precise arithmetic criterion for irre…

Merino's conjecture. This curve is the union of a finite number of Lissajous curves. The conjecture concerns the decomposition of the Chebyshev-defined curve into parametrized Liss…

Unit-primitive matrix conjecture. The characteristic polynomial of $A_n$ is

Let $\kappa\geq2$ be even and fix $1\leq j\leq\lfloor\kappa/2\rfloor$. Let $T_\kappa$ be the Chebyshev polynomial used in the paper, and choose real numbers satisfying … with…

Let $(r_k(n))_{k\geq 0}$ be the companion sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the relevant dilated Chebyshev value. Prime…

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer $n$. For a fixed $n$, define $(k_N)_{N\geq 1}$ by requiring that $s_{k_N}(n)$ is the $N$th prime term…

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the dilated Chebyshev polynomial of the first kind ev…

Let $B_{\alpha,n}$ be the rectangular band matrix defined by … for $i\le k\le n-1+i$, with zero entries otherwise. A matrix is totally positive if all of its minors are non-negativ…

Let $B_{\alpha,n}$ be the $n\times(2n-1)$ band matrix whose coefficients are … when $i\le k\le n-1+i$, and are zero otherwise. Let $B_\sigma(1)$ denote the associated coefficient i…

Let $\theta$ be a root of $\theta$'s defining polynomial … . Index conjecture. If $p eqell$ is an odd prime for which $tequivpm2\pmod{p^2}$, then … This conjectural formula is inten…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. For an odd integer $i\ge 3$, the difference-two crossing-number conjecture. The parametrization … parametriz…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. For an odd integer $i\ge 3$, the consecutive-degree crossing-number conjecture. The parametrization … parame…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. Let $i,j$ be relatively prime positive integers, and let $k,\ell$ belong to the semigroup denoted by…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind, and let $(i,j,k)$ be a reduced triple as in the source. Primality conjecture. Every knot of the form … for t…