4 problems

Three-rational-incidence saving conjecture. The number of such solutions is

Let $L$ be the linear functional in the preceding generalization, with integer parameters $\alpha$, $\beta_i$, and $\gamma_i$, and let $P_n/Q_n$ be the associated rational approximant…

Prescribed-prime-divisors conjecture. There is a rational number $A/Q$ such that $Q\leq X^{1+\epsilon}$, $\omega(Q)=n$, and

Sums of rational approximations conjecture. There are integers $a_1,\ldots,a_n$ and distinct primes $q_1,\ldots,q_n$ with $1\leq q_i\leq N$ such that