7 problems

Multivariable Bateman–Horn conjecture. As $P\to+\infty$,

Let $f_1(x_1,\ldots,x_k),\ldots,f_s(x_1,\ldots,x_k)$ be multivariable polynomials with integral coefficients. Write $Z_m^*=\{x\in\mathbb N\mid \gcd(x,m)=1,\ x\leq m\}$. Assume that…

Let $f(x_1,\ldots,x_k)$ be a multivariable polynomial with integral coefficients. Write $Z_m^*=\{x\in\mathbb N\mid \gcd(x,m)=1,\ x\leq m\}$. Assume that there is a positive integer…

Let $f_1(x_1,\ldots,x_k),\ldots,f_s(x_1,\ldots,x_k)$ be integral linear polynomials satisfying the preceding local condition with constant $c$. Simultaneous-prime conjecture for li…

Let $f_1(x_1,\ldots,x_k),\ldots,f_s(x_1,\ldots,x_k)$ be multivariable polynomials with integral coefficients. Assume that there is a positive integer $c$ such that for every intege…

Conjecture 2. Suppose there is a positive integer $c$ such that, for every positive integer $m\geq c$, some integral point $(y_1,\ldots,y_k)$ makes all values…

Conjecture 1. If these functions represent simultaneous primes for infinitely many integral points $(x_1,\ldots,x_k)$, then there is a constant $c$ such that for every positive int…