4 problems

Prime-power congruence conjecture. One has

Modular-3 conjecture. For every $k$,

The prime-power generalization conjecture. For every integer $k\geq0$, there exists such a polynomial $c_k(t)$ for which

Let $n \gg k > 2$, and let $\lambda_1,\ldots,\lambda_k \in \mathbb{Q}$ satisfy $\lambda_i>0$ for each $i$ and … Set $n_i=\lambda_i n$ for each $i$, so that the $\lambda_i$ determin…