Browse Problems

The **Beal Conjecture**: if we are given positive integers $A, B, C, x, y, z$ such that $x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ …

The **Generalized Riemann Hypothesis** asserts that all the non-trivial zeros of the Dirichlet $L$-function $L(\chi, s)$ of a primitive Dirichlet character $\chi$ have real part $\frac{1}{2}$.

**NP ≠ coNP**: The conjecture that the complexity classes NP and coNP are not equal.

**P ≠ NP**: The conjecture that the complexity classes P and NP are not equal.

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 **Collatz conjecture** states that for any …

Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

Show that $r_k(N) = o_k(N / \log N)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression.

Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

If $A \subset \mathbb{N} has $\sum_{n \in A}\frac 1 n = \infty$, then must $A$ contain arbitrarily long arithmetic progressions?

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 $$ N \gg 2 ^ n. $$

Does this imply that $$ \liminf \frac{|A \cap [1,x]|}{x} = 0? $$

Or $$ \sum_{x \in A} \frac{1}{x \log x} < \infty, $$

Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$ [Er56] Erdős, P., _Problems and results in additive number theory_. Colloque sur la Th\'{E}orie des Nombres, Bruxelles, 1955 (1956), …

A number of improvements of the constant $\frac{1}{4}$ have been given, with the current record $\sqrt{2 / \pi}$ first provied in unpublished work of Elkies and Gleason.

The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$. https://oeis.org/A276661

The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$. https://oeis.org/A276661

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] …

If $A ⊆ \mathbb{N}$ is such that $A + A$ contains all but finitely many integers then $\limsup 1_A ∗ 1_A(n) = \infty$.

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). …

Erdős proved the following pairwise version. Let `A ⊆ ℕ` be an infinite set such that the pairwise sums `a + b` are all distinct for `a, b` in `A` (aside from …

Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg \frac{n}{\sqrt{\log n}}$ many distinct distances?

Guth and Katz [GuKa15] proved that there are always $\gg \frac{n}{\log n}$ many distinct distances. [GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erdős distinct distances problem in the plane. Ann. …

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $|A| = n$ then $\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors. Is it true that $\frac{f(n)}{\log n} \to\infty$?

Erdős and Turán proved [ErTu34] in their first joint paper that $$ \log n \ll f(n) \ll \frac{n}{\log n} $$ [ErTu34] Erdős, Paul and Turan, Paul, _On a Problem in the Elementary …

Erdős says that $f(n) = o(\frac{n}{\log n})$ has never been proved.

Theorem 5 of [BS2013]: when elliptic curves over ℚ are ordered by height, a density of at least 20.62% have rank 0.

Theorem 4 of [BS2013]: when elliptic curves over ℚ are ordered by height, a density of at least 83.75% have rank 0 or 1.

For every positive real number `ε`, there exist only finitely many triples `(a, b, c)` of coprime positive integers, with `a + b = c`, such that `c > rad(abc)^(1+ε)`

For every positive real number ε, there exists a constant `K_ε` such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have `c …

For every positive real number ε, there exist only finitely many triples `(a, b, c)` of coprime positive integers with `a + b = c` such that `q(a, b, c) > 1 …

An abelian group is a Leinster group if and only if it is cyclic with order equal to a perfect number. Reference: Leinster, Tom (2001). "Perfect numbers and groups". Theorem 2.1.

Conjecture: a(n) = 3^n + 3 * 2^n + 6 for n ≥ 1.

The **Agoh-Giuga Conjecture**, Agoh's formulation

Giuga showed that a counterexample Giuga number has at least 1000 digits. Ref: G. Giuga, _Su una presumibile proprieta caratteristica dei numeri primi_

Borwein, Borwein, Borwein and Girgensohn showed that any strong Giuga number has at least 13000 digits. Ref: D. Borwein, J. M. Borwein, P. B. Borwein, and R. Girgensohn, _Giuga’s conjecture on primality_

Bedocchi showed that any Giuga number has at least 1700 digits. Ref: E. Bedocchi, _Note on a conjecture about prime numbers_

The two statements of the conjecture are equivalent.

The **Agoh-Giuga Conjecture**, Giuga's formulation

Let `G(X)` denote the number of exceptions `n ≤ X` to Giuga’s conjecture. Then for `X` larger than an absolute constant which can be made explicit, `G(X) ≪ X^{1/2} log X`. Ref: …

Every strong Giuga number is a Carmichael number.

Giuga showed that a Giuga number has at least 9 prime factors. Ref: G. Giuga, _Su una presumibile proprieta caratteristica dei numeri primi_

Conjecture: the sequence A228828 is infinite.

Conjecture: $a(n) = O(\log(n)\log(\log(n)))$.

Are $e$ and $\pi$ algebraically independent?

Are all Fermat numbers are square-free?

The **Amitsur Conjecture**: If `J` is a nil ideal in `R`, then `J[x]` is a nil ideal of the polynomial ring `R[x]`. This is known to be false, see Agata Smoktunowicz, _Polynomial …

**Andrica's conjecture** The inequality $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.

Ferreira proved that Andrica's conjecture is true for sufficiently large n.