Open-problem statements, with their sources attached.

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 $5$, $S(k) > k$.

For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.

This conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.

TODO(lezeau): Formalize the Turing Machine version of this problem.

Source: Hardness of Busy Beaver Value BB(15): https://link.springer.com/chapter/10.1007/978-3-031-72621-7_9 This is also https://arxiv.org/abs/2107.12475.

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

If $A\subseteq\{1, ..., N\}$ with $|A| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then

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

A generalisation of the problem to sets $A \subseteq (0, N]$ of real numbers, such that the subset sums all differ by at least $1$ is proposed in [Er73] and [ErGr80].

[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

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

Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$?

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.

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers.

Ref: Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$

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 $|\alpha| < 1$)

Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).

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 $\mathbb{K}$.