84 problems

Goldbach-event independence conjecture. The events $A$ and $A'$ satisfy

Generalized Andrews identity.

Let $k\geq 9$, and let the $n$-th non-$k$-gonal pyramidal number be the $n$-th positive integer that is not a $k$-gonal pyramidal number. The preceding formula defines this sequenc…

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

Chen's conjecture. If $r_1,\ldots,r_t$ are positive integers with $r_1^{-1}+\cdots+r_t^{-1}\geq 1$, then $\mathcal{R}_{\mathcal{A}}\cap(\mathcal{R}_{\mathcal{A}}+2)$ has positive l…

Let $d\geq 2$ and $k\geq 1$. Let $A_i$ be positive integers, let $u_i,v_i$ be integers, let $\lambda\ne 0$ be a real constant, and let $L(M)$ be an integer-valued linear polynomial…

Let $d\geq 2$, $k\geq 1$, and $\varepsilon>0$. For $N\geq 1$ and $\beta\in\mathbb R/\mathbb Z$, consider the distance to the nearest integer, denoted by $\|\cdot\|$. Uniform expone…

For a fixed base $b\geq 2$, let $\overleftarrow{K}_b$ be the smallest number greater than $1$ such that every sufficiently large integer is a sum of at most $\overleftarrow{K}_b$ r…

For a fixed base $b\geq 2$, define the Nnamlerinchs constant $\overleftarrow{K}_b$ to be the smallest number greater than $1$ such that every sufficiently large integer can be writ…

Work in base $b=10$, and write $\overleftarrow{p}$ for the digit reversal of a prime $p$. Base-10 ternary reversed-prime conjecture. Every sufficiently large odd integer $N$ can be…

Let $b$ be a fixed prime, and let $\overleftarrow{p}$ denote the digit reversal of a prime $p$ in base $b$. Two-reversed-prime conjecture. Every sufficiently large even integer $N$…

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 $p_n$ denote a prime number, and for each integer $k\geq 1$ define … where $p_{n+i}$ are consecutive primes. Infinite admissible lengths conjecture. For every prime number…

Let $n$ be a natural number greater than $23$. A prime power is a number $P^k$ where $P$ is prime and $k\geq 2$ is an integer. Main conjecture on limited prime power sums. Every na…

Let $cmathbf a=(a_1,\cdots,a_k)$ be a positive integral vector of dimension $k>1$, and let $f(cmathbf x)=cmathbf acdotcmathbf x$ be its linear form. Let $A_n$ be the anti-recurrence…

An anti-recurrence sequence is the increasing sequence generated by a positive integral linear form, by selecting the integers represented by that form without repetition. A linear…

Generalization of Goldbach's conjecture. Let $Q$ be a subset of natural numbers whose distribution is similar to primes. There exists $N_Q\in\mathbb{N}$ such that, for any even int…

Minimal-order conjecture. As $n\to\infty$,

Let $q$ be a prime power and let $n$ satisfy the normal conditions for the Restricted Problem. An element $a\in\mathbb{F}_q$ is 5-potent if it is a fifth power, and $n$-potent if i…

Let $N>24476$ be a non-square satisfying $N\equiv 0,2\pmod{6}$. Nonsquare prime-square representation conjecture. Then there exist primes $p$ and $q$ such that … By Hua's theorem,…

Let $k\geq 1$ be odd, and let $N$ be a sufficiently large even number. Prime-power plus almost-prime conjecture. Then $N$ can be expressed as the sum of a $k$th power of a prime an…

Let $\widetilde{M}(k)$ denote the parameter in Theorem. Conjecture on $\widetilde{M}(k)$. One can take $\widetilde{M}(k)=k$ for all $k\geq 1$ in Theorem. This would improve the pap…

Signed pentagonal universality conjecture. For each $r\in\{\pm1,\pm3\}$, every integer $n>N(r)$ can be written as

Let $\mathbb N$ denote the nonnegative integers. Four-term pentagonal representation conjecture. Every integer $n>51$ can be written as … with $w,x,y,z\in\mathbb N$. This is an exp…

Let $\mathbb N$ denote the nonnegative integers and let $x,y,z\in\mathbb Z$. Exceptional-set conjecture. The following representation sets are exactly: … … … … These claims give pr…