21 problems

Irrationality conjecture for odd zeta values. For every $n\in\mathbb{N}$, $\zeta(2n+1)$ is irrational.

Multiplicativity conjecture. Measures of irrationality for complete intersections in projective space should behave multiplicatively in the degrees $d_1,\ldots,d_c$ of the defining…

Polynomial arctanh integral conjecture. For every nonzero integer polynomial $P$, one has

Product-of-curves conjecture.

Stapleton's asymptotic conjecture. The degree of irrationality of a very general polarized $K3$ surface grows on the order of $\sqrt d$. This refines the known upper bound and is c…

Let … be the hypergeometric value used in the paper. Irrationality conjecture. … This mild irrationality assumption is introduced for class 8, where the relevant even twist is not…

Theodorus studied the irrational square roots represented by the successive sides of right triangles in his spiral, including the square roots of $3,5,\ldots,17$. Theodorus's histo…

Stolarsky's conjecture. The series cannot be rational for every positive integer $t$.

Let $X\subseteq\mathbb{P}^{n+r}$ be a very general smooth complete intersection of dimension $n>0$, cut out by polynomials of degrees $d_{1},\ldots,d_{r}$, and let…

Let $\sigma_1(n)=\sum_{d\mid n}d$ be the sum of the divisors of $n$, let $\varphi(n)$ denote the Euler totient function, and let $\omega(n)$ denote the number of distinct prime div…

Erdős–Kac conjecture. The number $\alpha_k$ is irrational for every positive integer $k$. The conjecture is known unconditionally for $k\leq 3$, and the paper proves it for $k=4$;…

Let the sum run over all square-free positive integers $n$. Erdős's conjecture. The number … is irrational. This conjecture concerns irrationality phenomena for series whose expone…

The $\pi$–$e$ counting conjecture. As $x\to\infty$,

The Riemann zeta function is denoted by $\zeta$, and $\zeta(5)$ is its value at $5$; $1/\zeta(5)$ denotes its reciprocal. Irrationality conjecture. The real numbers $\zeta(5)$ and…

Let $X=X_d\subseteq\mathbf{P}^{n+1}$ be a very general smooth hypersurface of dimension $n\geq 2$ and degree $d\geq 2n+1$. Its degree of irrationality satisfies … Moreover, if…

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound by $1$. … Numerical data in the source suggest this stronger bound than the preceding conjecture. The…

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound. … This conjecture is motivated by estimating the corresponding series through the quantities…

Euler's constant irrationality conjecture. Euler's constant is irrational. This is a long-standing open problem concerning the arithmetic nature of Euler's constant.

Apéry-sequence Brun criterion conjecture. There is an unbounded subsequence of positive integers $n_k\to\infty$ such that $\delta_{n_k}<0$. The sequence $y_n/x_n$ is increasing and…

Chowla's conjecture. The values $f(x)$ and $g(x)$ are irrational at all such rational values of $x$.

Irrationality conjecture for $\alpha_{**}$. The number $\alpha_{**}$ is irrational.