7 problems

The rational approximation conjecture. For each $j=1,2,3,4$, one has

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…

Mainardi's bounding conjecture. For every $t>0$ and every $\alpha\in(0,1)$, these two rational functions provide upper and lower bounds for $e_\alpha(t)$.

Conjecture. Every positive irrational number has an E rational approximation. This conjecture proposes that every positive irrational admits a rational series expansion satisfying…

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