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Let $G$ be a finite group and $k$ a field of characteristic $p>0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_0$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_0$ has the property that $e_0{\left(k_P\right)}^G$ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_P$ is the trivial $kP$-module, ${\left(k_P\right)}^G$ is the induced module, and $e_0$ is the block idempotent of $B_0$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more than three...

In this paper, we find all Pell and Pell-Lucas numbers written in the form $-2^a-3^b+5^c$, in nonnegative integers $a$, $b$, $c$, with $0\le max\{a,b\}\le c$.