Browse Problems

Gauss proved that $$ |E(r)|\leq 2\sqrt{2}\pi r, $$ for sufficiently large $r$. [Ha59] Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). New York: Chelsea …

Hardy and Laundau independently found a lower bound by showing that $$ |E(r)| \neq o\left(r^{1/2}(\log r)^{1/4}\right) $$

Erdős and Szekeres proved the bounds $$ 2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1 $$ ([ErSz60] and [ErSz35] respectively). [ErSz60] Erdős, P. and Szekeres, G., _On some extremum problems in …

It is not known whether every Euclid number is a square-free number.

Euler's sum of powers conjecture states that for integers $n > 1$ and $k > 1$, if the sum of $n$ positive integers each raised to the $k$-th power equals another integer …

All unitary perfect numbers are even.

The value of $N(r)$ can be expressed as $$ N(r) = 1 + 4\sum_{i=0}^{\infty}\left(\left\lfloor\frac{r^2}{4i+1}\right\rfloor - \left\lfloor\frac{r^2}{4i + 3}\right\rfloor\right). $$

**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid |f(z)| < …

There exists a set of infinite $\mathrm{VC}_n$ dimension in $\mathbb R^{n + 2}$.

For every $n$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$.

There exists a set in $\mathbb R^3$ shattering an infinite set.

It is known that every Jordan curve admits at least one inscribed rectangle.

It is known that every *smooth* Jordan curve admits inscribed rectangles of all aspect ratios.

It is also known that every $C^2$ Jordan curve admits an inscribed square.

At least one of $\zeta(5), \zeta(7), \zeta(9)$ or $\zeta(11)$ is irrational. [Zu01] W. Zudilin (2001). _One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational_. Russ. Math. Surv. 56 (4): 774–776.

A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number. It is conjectured that there is at least one Wall–Sun–Sun …

Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime.

The expression $1 + n^{0.74}$ does not work as an upper bound.

A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that $nk + 1$ is a prime.

Does there exist a 3 by 3 matrix such that every entry is a distinct square, and all rows, columns, and diagonals add up to the same value?

If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing …

Non-abelian Leinster groups exist. For example, `S₃ × C₅` (order 30) and `A₅ × C₁₅₁₂₈` are Leinster groups. Reference: Leinster, Tom (2001). "Perfect numbers and groups".

Non-primitive terms have the form $m \cdot s$ where $m$ is primitive and $s$ is squarefree with $\gcd(m, s) = 1$.

There exists a semiring with a unique left maximal ideal but more than one right maximal ideals.

There exists a semiring with a unique left maximal ideal and a unique right maximal ideal which are not the same as sets.

For any $k$ there exists some $t_k\in (0,1)$ such that the set of $\alpha$ such that the sequence $\lfloor t_k\alpha^n\rfloor$ is complete consists of at least $k$ disjoint line segments.

Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact space X such that $𝔠 < |X|$.

$e + \pi$ is transcendental.

$e^e$ is transcendental.

$e\pi$ is transcendental.

$e^{\pi^2}$ is transcendental.

There are no distinct primes $p$ and $q$ such that $\frac{q^p - 1}{q - 1}$ divides $\frac{p^q - 1}{p - 1}$

The **Fermat–Catalan conjecture** states that the equation $a^m + b^n = c^k$ has only finitely many solutions $(a,b,c,m,n,k)$ with distinct triplets of values $(a^m, b^n, c^k)$ where $a, b, c$ are positive …

By the **Darmon-Granville** theorem, for any fixed choice of positive integers m, n and k satisfying $\frac 1 m + \frac 1 n + \frac 1 k < 1$, only finitely many …

Are Fermat numbers composite for all `n > 4`?

There are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime It is also a barrier to defining a benchmark from this paper: https://arxiv.org/html/2505.13938v1 (see Figure 8).

There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime.

Note that this is a stronger form of `Equation255_not_implies_Equation677`.

The negation of `Finite.Equation677_not_implies_Equation255`. Probably this is false.

The negation of `Finite.Equation677_implies_Equation255`. Probably this is true. It would be a stronger form of `Equation677_not_implies_Equation255`. Discussion thread here: https://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/FINITE.3A.20677.20-.3E.20255

From [PPVW2016], Section 8.2: "Our heuristic predicts (a) All but finitely many E ∈ ℰ satisfy rk E(ℚ) ≤ 21". In other words, there are only finitely many elliptic curves over ℚ …

**Firoozbakht's conjecture** The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$.

The inequality $p_{n+1}-p_n < (\log p_n)^2-\log p_n$ holds for all $n>4$. A consequence of Firuzbakht's conjecture.

Problem 15 in [Ar2013]: Is every homogeneous ω-monolithic compact space first countable?

Let $P = (m_1, \dots, m_k)$ be a tuple of positive even integers. Let $\pi_P(n)$ denote the number of primes $p\leq n$ such that $(p, p + m_1, \dots, p + m_k)$ …