18 problems

Let $r_2'(N)$ denote the number of representations of $N$ as a sum of two primes, and let $\mathfrak{S}(N)$ be the singular series … For even integers $N$, let $r_2(N)$ denote the…

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…

Pure reversed-prime asymptotic conjecture. For every $A>0$,

Let $b\geq 2$ be fixed. Define … and … Also let $\#\mathfrak{B}(N)$ denote the number of admissible integer representations encoded by the source's set $\mathfrak{B}(N)$. Asymptoti…

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 $1<c<d$ be relatively prime integers and set … Let $\pi_{c,d}$ denote the number of primes not exceeding $g_{c,d}$ that can be written as $cx+dy$ with…

Generalised Goldbach conjecture. Under these conditions, there exist primes $p$ and $q$ such that

Iwaniec's conjecture. The limit exists and equals

Let $c_0>1$. For fixed $c$ with $1<c<c_0$ and every sufficiently large positive integer $N$, consider the equation … where $p_1,p_2,p_3$ are prime numbers and each $x_i,y_i$ is an…

Bounded odd Gaussian-prime representation conjecture. There are constants $c_0$ and $k_0$ such that, for every $z\in\Gamma_G$ with…

Finiteness conjecture. The sequence $s(n)$ contains only finitely many zero terms. This asserts that $p_n+k-3$ is represented by $F_k$ for some $k\geq3$ for all but finitely many p…

Let $R(n)$ denote the number of ordered pairs of odd primes $(p,q)$ satisfying $p+q=n$, and let … be the twin-prime constant. Hardy–Littlewood's conjecture. As $n$ tends to infinit…

Let $F_n$ denote the Fibonacci sequence. A Fibonacci number is positive when its value is positive, and an odd Fibonacci number is one not divisible by $2$. Sun's prime–Fibonacci-p…

Let $P_0=0$, $P_1=1$, and $P_{n+1}=2P_n+P_{n-1}$ for $n=1,2,3,\ldots$ be the Pell sequence. Sun's prime–Pell-number conjecture. Every integer $n>5$ can be written as the sum of an…

Let $F_0=0$, $F_1=1$, and $F_{i+1}=F_i+F_{i-1}$ for $i=1,2,3,\ldots$ be the Fibonacci sequence. A Fibonacci number is positive when its index is positive and its value is positive.…