15 problems

Let $L \in \text{\bbold Q}[z,E]$ have leading and trailing coefficients $a_d(z)$ and $a_0(d)$. Let $N_1$ be an integer larger than any integer root of $a_d(z-d)a_0(z)$, and let…

Arithmetic inclusion conjecture. There holds the inclusion

Let an arithmetic order be a subring of a finite-dimensional semisimple algebra over a number field, and let a lattice over that order be a finitely generated torsion-free module.…

Let … psiA:mathbb{F}p(x)^ntomathbb{F}p(x)^n, qquad Y(x)mapsto(partial-Ap)^pY(x). … admits a basis of algebraic solutions if and only if its $p$-curvature vanishes for almost all, t…

The paper studies the weak regularity principle $WR(\Sigma_{n+1}\wedge\Pi_{n+1})$ and its relationship with induction: for each $n\in\mathbb{N}$, this principle is equivalent to…

Let $F$ be a number field with ring of integers $\mathcal{O}_{F}$, and consider indefinite quadratic forms over $\mathcal{O}_{F}$. Xu's arithmetic Springer conjecture. The arithmet…

Let $\mathscr{L}\in\mathbb{Q}(x)\langle\partial_x\rangle$ be the differential operator attached to the paper's linear differential equation, and let $\mathscr{L}_p$ denote its redu…

Induction-with-inequality hypothesis. The introduced structure $d4c^+$ is a model of $d400(\ne)$.

Characterizations and reductions conjecture. The following characterizations and reductions hold:

Let $T$ be a theory, and let ${\mathrm{RC}^\nabla\hspace{-0.7pt}}$ be the reflection calculus with modalities $\Diamond_n$ and $\nabla_n$. An arithmetical interpretation $*$ in…

Let $\mathcal B$ be a model of $\mathrm{I}\Sigma_1$. Every element of $\mathcal B$ has a unique prime factorization, and $\operatorname{rad}(a)$ denotes the product of the distinct…

Fermat's Last Theorem is an arithmetical statement concerning solutions of the equation $x^n+y^n=z^n$ in positive integers for exponents $n>2$. Friedman's conjecture. Fermat's Last…

Consider the displayed numerical equality. Equality claim. The following equality holds: … This is an elementary arithmetic identity and is therefore solved.

Conjecture.

Let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, and consider a scalar modular form for a subgroup of $\Gamma$. Say that it has bounded denominator when its Fourier coefficients have…