Browse Problems

For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation $ax^n - by^m = c$ where $(m, n) \neq (2, 2)$ …

$\pi^e$ is transcendental.

$\pi^{\pi^{\pi^\pi}}$ is not an integer. This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation. …

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

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

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

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

**Polignac's conjecture** For any integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime.

Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most $5$ tetrahedral numbers.

Salzer–Levine strengthening (as stated on Wikipedia/OEIS): there are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$.

For members of the sequence other than $8$, we have $k + 1$ is prime.

For any `k ≥ 2`, let `a₁,...,aₖ` and `b₁,...,bₖ` be integers with `aᵢ > 0`. Suppose that for every prime `p` there exists an integer `n` such that `p ∤ ∏ i, …

[PPVW2016] 8.2(b): for 1 ≤ r ≤ 20, the number of elliptic curves over ℚ with rank `r` and naïve height at most `H` is asymptotically `H ^ ((21 - r) / …

Does there exist a point in the plane at rational distance from all four vertices of the unit square?

Can a unit square be covered by rectangles of width `1 / (n + 1)` and height `1 / (n + 2)`?

It is known that packing the rectangles into a square of side length `133/132` is possible. Reference: https://www.sciencedirect.com/science/article/pii/0097316594901163

It is known that packing the rectangles into a square of side length `501/500` is possible. Reference: https://www.sciencedirect.com/science/article/pii/S0167506008706009

Equivalently, can a unit square be packed with rectangles of width `1 / (n + 1)` and height `1 / (n + 2)`?

Conjecture: The set of regular primes is infinite.

$\log(\pi)$ is transcendental.

$\log(\log(2))$ is transcendental.

**Same parity betrothed numbers conjecture.** Do there exist betrothed numbers $(m, n)$ where both have the same parity (both even or both odd)? All known betrothed pairs consist of one even and …

Given any set of $n$ complex numbers $\{z_1, ..., z_n\}$ that are linearly independent over $\mathbb{Q}$, the field extension $\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n})$ has transcendence degree at least $n$ over …

**Schinzel conjecture (H hypothesis)** If a finite set of polynomials $f_i$ satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers $n$ such that $f_i(n)$ are primes for all $i$.

**Schur's Theorem (1924):** Let `f_n(x) = ∑_{j=0}^n x^j/j!` be the `n`-th truncated exponential polynomial over `ℚ`. Then for `n ≥ 2`: - If `n ≡ 0 (mod 4)`, the Galois group of …

For integers $x, y \geq 2$, $$ \pi(x + y) \leq \pi(x) + \pi(y), $$ where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$.

**PSW conjecture** (Selfridge's test) Let $p$ be an odd number, with $p \equiv \pm 2 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p+1} \equiv 0 \pmod{p}$, then $p$ is a prime number.

Selfridge conjectured that the number of prime factors of the `n`-th Fermat number does not grow monotonically in $n$.

Selfridge conjectured that the number of prime factors of the `n`-th Fermat number does not grow monotonically in $n$. A sufficient condition for this conjecture to hold is that there exists a …

**Sendov's conjecture** states that for a polynomial $$f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at …

**Sendov's conjecture** states that for a polynomial $$f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at …

**Sendov's conjecture** states that for a polynomial $$f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at …

$\sin(e)$ is transcendental.

The longest snake in the $0$-dimensional cube, i.e. the cube consisting of one point, is zero, since there only is one induced path and it is of length zero. -/ @[category test, …

The longest snake in the $0$-dimensional cube, i.e. the cube consisting of one point, is zero, since there only is one induced path and it is of length zero. -/ @[category test, …

The longest snake in the $0$-dimensional cube, i.e. the cube consisting of one point, is zero, since there only is one induced path and it is of length zero. -/ @[category test, …

An upper bound of the maximal length of the longest snake in a box is given by $$ 1 + 2^{n-1}\frac{6n}{6n + \frac{1}{6\sqrt{6}}n^{\frac 1 2} - 7}. $$

Strong Sensitivity Conjecture, for every Boolean function `f : {0,1}^n → {0,1}`, `bs(f) ≤ s(f)^2`. We call this the *strong* sensitivity conjecture because the original sensitivity conjecture only asked for a polynomial …

Suk [Su17] proved $$ f(n) ≤ 2^{(1+o(1))n}. $$ [Su17] Suk, Andrew, _On the Erdős-Szekeres convex polygon problem_. J. Amer. Math. Soc. (2017), 1047-1053.

The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$ for the sum of squares of gaps between consecutive primes.

Sylvester and Schur: for $1 \le i \le n/2$ there is a prime $p > i$ dividing `n.choose i`.

Can every odd integer greater than 5 be written as the sum of three primes? (A prime may be used more than once.) NB. While Harald Helfgott's solution is not published in …

Let $\{S_k\}$ be any sequence of sets in the complex plane, each of which has no finite limit point. Then there exists a sequence $\{n_k\}$ of positive integers and a transcendental entire …

**Theorem 1.** If a triangle-free graph has `f = 2`, then it is bipartite, has minimum degree `1`, and its degree sequence is compact.

**Theorem 2.** Every triangle-free graph is an induced subgraph of one with `f = 3`.

The Catalan constant $G$ is transcendental.