Conjectures on representations involving primes

Abstract: We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists $k\in{0,\ldots,n}$ such that $n+k$ and $n+k^2$ are both prime. (ii) Each integer $n>1$ can be written as $x+y$ with $x,y\in{1,2,3,\ldots}$ such that $x+ny$ and $x^2+ny^2$ are both prime. (iii) For any rational number $r>0$, there are distinct primes $q_1,\ldots,q_k$ with $r=\sum_{j=1}^k1/(q_j-1)$. (iv) Every $n=4,5,\ldots$ can be written as $p+q$, where $p$ is a prime with $p-1$ and $p+1$ both practical, and $q$ is either prime or practical. (v) Any positive rational number can be written as $m/n$, where $m$ and $n$ are positive integers with $p_m+p_n$ a square (or $\pi(m)\pi(n)$ a positive square), $p_k$ is the $k$-th prime and $\pi(x)$ is the prime-counting function.