209 problems

Let $n$ be an odd integer greater than $3$. De Polignac's conjecture. Every such $n$ can be written as … where $p$ is a prime and $a$ is a nonnegative integer. This is a classical…

Let $n$ be an odd integer with $n\geq 7$. Ternary Goldbach conjecture. Every such $n$ can be written as a sum of three primes. This conjecture is stated as background for Sárközy's…

Cilleruelo's conjecture. For any $r\ge 2$,

An even integer is called a Goldbach number if it can be written as a sum of two primes. Binary Goldbach conjecture. Every even integer greater than $2$ is the sum of two primes. T…

For each $k\in\mathbb{N}$, let $g(k)$ be the smallest number such that every positive integer is a sum of $g(k)$ non-negative $k$-th powers. Waring's conjectural formula. It is con…

Let $n$ be an odd integer greater than $7$, and let $p$, $q$, and $r$ denote prime numbers. The weak Goldbach conjecture. Every such $n$ can be written as … The source describes ex…

Let $\mathbb{P}$ denote the set of primes. An even number is an element of $2\mathbb{N}$. Kronecker's conjecture. Every even number can be expressed in infinitely many ways as the…

Let $\mathbb{P}$ denote the set of primes. An even number is an element of $2\mathbb{N}$. Maillet's conjecture. Every even number is the difference of two primes. This is a foundat…

A cover of the integers is a finite disjoint collection of residue classes … For a non-trivial cover, let $d_s$ be the largest index among the moduli $d_1,\ldots,d_s$. Erdős's conj…

Sun's conjecture. Every natural number other than $216$ can be written as

Let $C$ be the classical Cantor ternary set, and let $x_1,x_2,x_3,x_4$ be elements of $C$. Athreya–Reznick–Tyson's four-squares conjecture. Every element of $[0,1]$ can be expresse…

An odd integer is an integer not divisible by $2$, and a prime is an integer greater than $1$ with no positive divisors other than $1$ and itself. Lemoine's conjecture. Any odd int…

Let $n$ be an even integer greater than $2$, and let $p$ and $q$ denote prime numbers. The strong Goldbach conjecture. Every such $n$ can be written as … The source reports that th…

Let $\mathcal{P}$ denote the set of prime numbers. A set of positive integers is totally primitive if it is not asymptotically additively decomposable: there do not exist sets…

Let $b\geq 2$ be a fixed base. For a positive integer, write $\overleftarrow{n}$ for its digit reversal in base $b$. Then $p_1$ and $p_2$ are primes, and…

Let $d\geq 3$ be an integer, let $N\in\mathbb N$, let $a_1,\ldots,a_N\in\mathbb C$, and write $\mathbb T^2$ for the two-dimensional torus. For $p\geq 2(d+1)$ and every…

Let $\mathscr{A}$ and $\mathscr{B}$ be two sets of positive integers. For a set $S$ of positive integers, write $S(x)=|S\cap[1,x]|$. Chen's Romanov-type conjecture. If there exists…

Pollock's conjecture. Every positive integer is the sum of at most five tetrahedral numbers, seven octahedral numbers, nine cubes, thirteen icosahedral numbers, and twenty-one dode…

Farhi's conjecture. Every natural number can be represented as the sum of three terms of this sequence.

Seven-cubes conjecture. Every integer $N>454$ is the sum of seven nonnegative cubes. This is an additive number theory problem concerning representations by sums of cubes. The supp…

Hua's conjecture. Every sufficiently large $n\in\mathcal H_2$ can be represented as the sum of a prime and the square of another prime, and every sufficiently large…

Equal-sums conjecture. The quantity $L_s(f;B)$ is dominated by the $s!B^s$ trivial solutions in which $x_1,\ldots,x_s$ is a permutation of $x_{s+1},\ldots,x_{2s}$. This is stated a…

The three-powers representation conjecture. As soon as $d\geqslant4$,

Consider the sequence of fifth powers … A Sidon sequence is a sequence in which all sums of two elements are distinct up to rearrangement of the summands. Fifth-power Sidon conject…

Density conjecture for sums of distinct powers. If every pair $a_1,a_2\in A$ satisfies $\gcd\{a_1,a_2\}=1$ and