1,330 problems

An old problem of mine states as follows: Let $a_1 < a_2 < \cdots < a_k$ be a Sidon sequence. Can one extend it to a larger Sidon sequence … In other words, loosely speaking: can o…

More generally: Let $a_1 < a_2 < \cdots < a_{k_1} \leq n$, $b_1 < b_2 < \cdots < b_{k_2} \leq n$ be two Sidon sequences for which $a_j - a_i \neq b_v - b_u$ for all $i < j$ and…

Let $1 \leq a_1 < a_2 < \cdots < a_h \leq n$ be a maximum Sidon sequence. Can one find a Sidon sequence $b_1 < b_2 < \cdots < b_r \leq n$ for every $r$ and $n > n_0(r)$ so that the…

Here is an old conjecture of mine: Let $a_1 < a_2 < \ldots$ be an infinite sequence for which all the triple sums $a_i + a_j + a_k$ are distinct. Is it then true that…

Sidon also asked: Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence for which all the sums $a_i + a_j$ are distinct. Put … Probably there is a Sidon sequence $A$ for which ……

A sequence A is called by Khintchin an essential component if for every sequence B with $d_s(\mathrm{B}) > 0$, $d_s(\mathrm{A} + \mathrm{B}) > d_s(\mathrm{B})$. [...] This led me t…

A somewhat similar question is the following one: Let $a_1, a_2, \ldots, a_{2n}$ be $2n$ integers in the interval $(1, 4n)$ and let $b_1, b_2, \ldots, b_{2n}$ be the other $2n$ int…

Sidon also asked: Let $1 \leq a_1 < a_2 < \cdots < a_k \leq n$ and assume that $a_i + a_j$ are all distinct. Put $h(n) = \max k$. Determine or estimate $h(n)$ as accurately as poss…

Now I state a few problems of Nathanson and myself : Let $a_1 < a_2 < \cdots$ be an infinite sequence of integers; denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Deno…

I conjectured long ago that if … then the $a_i$'s contain arbitrarily long arithmetic progressions. If true this of course implies that there are arbitrarily long arithmetic progre…

Let $1 \leq a_1 < a_2 < \cdots < a_k \leq n$ be a sequence of integers for which all the subset sums $\sum_{i=1}^{k} \varepsilon_i a_i$ ($\varepsilon_i = 0$ or $1$) are distinct. T…

For a finite set $A$ in a commutative ring, write … The sum-product problem asks how small these two sets can be simultaneously. Erdős–Szemerédi sum-product conjecture. For…

A natural number is called even if it is divisible by $2$, and a prime is a natural number greater than $1$ with no positive divisors other than $1$ and itself. Goldbach's conjectu…

Erdős–Turán conjecture. If $A$ is an asymptotic basis, then the representation function $f_A(n)$ cannot be bounded.

Erdős–Heilbronn conjecture. For every nonempty subset $A\subseteq {\mathbb Z}/p{\mathbb Z}$,

Let $A\subset\mathbb{F}_{2}^{n}$ be a set with $|A+A|\leqslant K|A|$. Polynomial Freiman-Ruzsa conjecture in characteristic two. Then $A$ is covered by at most $2K^{C}$ cosets of s…

Sárközy's conjecture. For all sufficiently large primes $p$, the quadratic residues modulo $p$ are additively irreducible.

Cameron–Erdős conjecture.

Cameron's conjecture.

Let $C_1$ and $C_2$ be Cantor sets in $\mathbb R$ with Lebesgue measure zero. Palis hypothesis. The arithmetic sum or difference $C_1\mathbin{+}C_2$ or $C_1\mathbin{-}C_2$ is eithe…

Let $G$ be an abelian group, and let $A=\{a_1,\ldots,a_k\}$ and $B=\{b_1,\ldots,b_k\}$ be subsets of $G$, where $G$ has odd order. Snevily's conjecture. There is a permutation…

Davenport constant conjecture for intervals.

Let $A$ be a finite set of integers, and write its mask polynomial as $A(X)=\sum_{a\in A}X^a$. Let $S_A$ be the set of prime powers $p^\alpha$ such that $\Phi_{p^\alpha}(X)$ divide…

Let $f(n)$ be the largest cardinality of a Sidon set contained in $\{1,2,\ldots,n\}$, where a subset $\mathcal{S}$ of the natural numbers is Sidon if all sums $s_1+s_2$ with…

Let $A\subseteq\mathbb{F}_2^n$ have doubling constant $K$, meaning that $|A+A|\leq K|A|$. A Freiman–Ruzsa theorem says that $A$ can be covered by $F(K)$ translates of a subspace…