Open-problem statements, with their sources attached.

There are infinitely many primes $p$ for which $p+2$ is also prime.

For every positive integer $n$, there is a prime $p$ satisfying $n^2<p<(n+1)^2$.

Every perfect number $n$, satisfying $n=\sum_{d\mid n,\ d<n}d$, is even.

The number $\pi$ is normal in base $10$, so every block of $k$ decimal digits has limiting frequency $10^{-k}$.

The sequence will eventually reach $1$.

If $n = ab$ is a crystal, then there are no other pairs of positive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and $B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique.

Conjecture 1.1*: For any odd prime $k$, the sum associated with the classical theta function $θ_3$, $S(k)$ is positive.

Conjecture 4.1*: For any prime $k$ larger than $5$, $S(k) > k$.

Conjecture 4.2*: For any prime $k$ larger than $233$, $S(k) > 2k$.

Conjecture 4.3*: For any prime $k$ larger than $3119$, $S(k) > 3k$.

Conjecture 4.4*: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$), $nk < S(k)$.

Let D be the diagonal group of SL_n(ℝ) where n ≥ 3. Then any relatively compact D-orbit in SL_n(ℝ) / SL_n(ℤ) is closed.

Problem 10.1. Are there a transcendental number $\alpha$ and a positive real number $\xi$ such that $\lVert \xi \alpha^n \rVert$ tends to~$0$ as~$n$ tends to infinity? [Har19] (Trivial for $|\alpha| < 1$)

Problem 10.2. To prove that $\lVert e^n \rVert$ does not tend to 0 as n tends to infinity.

Problem 10.3. To prove that there exists a positive real number~$c$ such that $\lVert e^n \rVert > e^{−cn}$, for every~$n \ge 1$. Posed by Mahler [Mah53].

Waldschmidt [Wal03] conjectured that a stronger result holds, namely that there exists a positive real number~$c$ such that $\lVert e^n \rVert > n^{-c}$ for every~$n \ge 2$. This is supported by metrical results [Kok45].

Note: the bound $n^{-c}$ equals $1$ when $n = 1$ for all $c$, while the distance to the nearest integer is always at most $1/2$, so the conjecture must start at $n \ge 2$.

Problem 10.4. Let $\xi$ be a non-zero real number and $\alpha > 1$ be a real number. The spectrum of the sequence $(\xi \alpha^n)_{n \ge 1}$ is at most countable. Posed by Mendès France [Men73].

Problem 10.5 (first part). Let $\mathbb{K}$ be a real number field. Then, for any $\varepsilon > 0$, there exists a lacunary sequence $(t_n)_{n \ge 1}$ of positive numbers in $\mathbb{K}$ such that $\limsup_{n \to \infty} \{\xi t_n\} \ge 1 - \varepsilon,$ for any real number $\xi$ not in $\mathbb{K}$.

Problem 10.5 ("moreover" clause). With the same hypotheses as problem_10_5, the sequence $(t_n)$ can be chosen so that, for any real $\xi$ not in $\mathbb{K}$, each subinterval of $[0, 1]$ of length $\varepsilon$ contains a limit point of the sequence $(\{\xi t_n\})_{n \ge 1}$. This is strictly stronger than problem_10_5: the limsup bound is the special case at the subinterval $[1 - \varepsilon, 1]$.

Problem 10.6. Find a very rapidly increasing sequence $(m_n)_{n \ge 1}$ of positive integers such that $(\{\xi m_n\})_{n \ge 1}$ is dense modulo one for every irrational number $\xi$. Note: Furstenberg's $2^m3^n$ is sublacunary but requires two parameters.