Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least $\mathcal{P}_2$ which satisfies $\mathcal{P}_2\equiv a\pmod q$. In this paper, it is proved for sufficiently large $q$, there holds \begin{equation*} \mathcal{P}_2(a,q)\ll q^{...

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ a non-negative integer, and almost but finitely many of them. We prove that for any $k,\ell$ such $k\nmid\ell$ there exists an diagonal ternary form. also conjecture are only primes $p$ which $(p,\ell)$-universal (for $\ell<p$) we show results computer experiments speak in favor conjecture.