Browse Problems

**The infinitude of Sophie Germain primes** There are infinitely many primes $p$ such that $2p + 1$ is prime.

**The infinitude of sexy primes** There are infinitely many primes $p$ such that $p + 6$ is prime.

The set of irregular primes is infinite.

**Inscribed rectangle problem** Does every Jordan curve admit inscribed rectangles of any given aspect ratio?

**Inscribed square problem** Does every Jordan curve admit an inscribed square?

Show that every bounded linear operator `T : H → H` on a separable Hilbert space `H` of dimension at least 2 has a non-trivial closed `T`-invariant subspace: a closed linear subspace …

Every (bounded) linear operator `T : H → H` on a finite-dimensional linear space `H` of dimension at least 2 has a non-trivial (closed) `T`-invariant subspace. This can be solved using the …

There exists a bounded linear operator `T` on the l1 space `(lp (fun (_ : ℕ) => ℂ) 1))` without non-trivial closed `T`-invariant subspace [Read 1985](https://doi.org/10.1112/blms/17.4.305), see also the first counterexample by …

Every bounded linear operator `T : H → H` on a non-separable Hilbert space `H` has a non-trivial closed `T`-invariant subspace. Such an invariant space is given by considering the closure of …

Every normal linear operator `T : H → H` on a Hilbert space `H` of dimension at least 2 has a non-trivial closed `T`-invariant subspace. If `T` is a multiple of the …

The **Inverse Galois Problem**: every finite group is isomorphic to the Galois group of a Galois extension of the rationals.

Every finite abelian group is realizable.

Every finite group is realisable over the field of rational functions with coefficients `K`, where `K` is any field of characteristic 0.

Every finite group is realisable over the field of rational functions with complex coefficients.

Every finite cyclic group is realizable.

Every finite symmetric group is realizable.

Is the Catalan constant $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$ irrational?

$\zeta(11)$ is irrational.

Is $e + \pi$ irrational?

Is $e \pi$ irrational?

Is $e ^ e$ irrational?

Is the Euler-Mascheroni constant $\gamma$ irrational?

$\zeta(5)$ is irrational.

Is $\ln(\pi)$ irrational?

$\zeta(9)$ is irrational.

$\zeta(2n + 1)$ is irrational for any $n\in\mathbb{N}^{+}$.

Is $\pi ^ e$ irrational?

Is $\pi ^ \pi$ irrational?

$\zeta(7)$ is irrational.

$\zeta(3)$ is irrational. [Ap79] Apéry, R. (1979). _Irrationalité de ζ(2) et ζ(3)_. Astérisque. 61: 11–13.

Find an accumulation point of the sequence `(3/2)^n` modulo `1`.

The sequence `(3/2)^n` has infinitely many accumulation points modulo `1`.

The sequence `(3/2)^n` is equidistributed modulo `1`.

Does the Rupert property hold for every convex polyhedron with nonempty interior?

Giuga showed that a number `n` is strong Giuga if and only if it is Carmichael and `∑_{p|n} 1/p - 1/n ∈ ℕ` (i.e., if and only if it is Carmichael and …

An integer `n : ℤ` can be written as a sum of three cubes (of integers) if and only if `n` is not `4` or `5` mod `9`.

Any rational number is a sum of three rational cubes. First proved by Ryley in 1825, which can be found in [Ri1930]. The below parametrization is brought from the MSE answer [MSE]. …

A composite number $n$ is weak Giuga if and only if $p \mid (\frac{n}{p} - 1)$ for all prime divisors $p$ of $n$.

A composite number $n$ is weak Giuga if and only if $$ \sum_{p\mid n} \frac{1}{p} - \frac{1}{n} \in\mathbb{N}. $$

The **Jacobian Conjecture**: any regular function (i.e. vector valued polynomial function from) `kⁿ → kᵐ` whose Jacobian is a non-zero constant has an inverse that is given by a regular function, where …

Is the chromatic number of `J(2 * k, k)` always at least `k + 2`?

Ma and Tang have proved that the chromatic number of $J(2k,k)$ is $>k+1$ for all $k>2$ not of the form $p-1$ for prime $p$.

It can be seen that the chromatic number of $J(2k,k)$ is $>k+1$ for all odd $k>2$.

The smallest case not on this page is $k=9$: But that one can be solved as well: The chromatic number of $J(18, 9)$ is at least $11$.

It is known that for $3 \leq k \leq 8$, the chromatic number of $J(2k, k)$ is greater than $k+1$, see [Johnson graphs](https://aeb.win.tue.nl/graphs/Johnson.html).

Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term. The **Juggler Conjecture** states that for any …

The two-dimensional case, proved by Davies [Da71]. [Da71] Davies, R. O., _Some remarks on the Kakeya problem_. Math. Proc. Cambridge Philos. Soc. 69 (1971), no. 3, 417–421.

The three-dimensional case, proved by Wang, Zahl [WaZa25]. [WaZa25] Wang, H. and Zahl, J., _Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions_. arXiv preprint, arXiv:2502.17655, …

The finite field Kakeya conjecture asserts that any Kakeya set in `𝔽_qⁿ` has size at least `c_n · qⁿ` for some constant `c_n` depending only on `n`. This was first proved by …