4 problems

Let $k\geq 1$ be odd, and let $N$ be a sufficiently large even number. Prime-power plus almost-prime conjecture. Then $N$ can be expressed as the sum of a $k$th power of a prime an…

Let $\widetilde{M}(k)$ denote the parameter in Theorem. Conjecture on $\widetilde{M}(k)$. One can take $\widetilde{M}(k)=k$ for all $k\geq 1$ in Theorem. This would improve the pap…

Let $\mathbb{P}_\ell$ denote the set of integers with the relevant almost-prime parameter $\ell$, let $\1_{\mathbb{P}_\ell}$ be its indicator, and let $\overline{\pi}_\ell(N)$ deno…

Constancy conjecture for $\Omega_k$ and $\Omega_k^{\mathrm{ext}}$. The function $\Omega_k(\vartheta,\vartheta_0)$ is constant with respect to $\vartheta_0$. The same applies to…