Open-problem statements, with their sources attached.

Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$?

Can infinitely many $n$ reach the same prime under the iteration $n\mapsto\phi(n) + 1$?

Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$?

What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto\phi(n) + 1$?

Conjecture 7 from Kahn–Kalai 2006: the same statement as the original conjecture, but with the additional assumption that $t$ is the critical probability for $F$, namely $\mu_t(F) = 1/2$.

How many iterations of $n\mapsto\sigma(n) - 1$ are needed before a prime is reached?

Special case in dimension $6$: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^6$.

If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ necessarily reaches a prime. Note: this is open , it is not clear that the σ iteration always terminates, since it is non-decreasing (unlike the φ iteration which is strictly decreasing).

Special case in dimension $10$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{10}$.

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. What is $\Theta(c(n))$?

Special case in dimension $12$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{12}$.

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

Special case in dimension $14$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{14}$.

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

Special case in dimension $15$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{15}$.

Is it true that iterates of $n\mapsto\sigma(n) - 1$ always reach a prime?

Open Quantum Problem 13: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^d$ for $d \ge 2$.

Let A ⊆ ℕ be an infinite set such that the triple sums a + b + c are all distinct for a, b, c in A (aside from the trivial coincidences). Is it true that liminf n → ∞ |A ∩ {1, …, N}| / N^(1/3) = 0?

Open benchmark statement: does an $\mathrm{AME}(7,6)$ state exist?

Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.

Is it true that $\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞$?

This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory", 1990).