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Source. Definition 5.2 and Theorem 5, printed p. 1747, physical PDF p. 9; the proof runs to printed p. 1748.

Convention

Definition 5.2, which the paper takes from Brunner, Caldwell, Krywaruczenko and Lownsdale: for a positive integer $b$, an integer $k>1$ is a $b$-Sierpiński number when $\gcd(k+1,b-1)=1$, $k$ is not a power of $b$, and $k\cdot b^n+1$ is composite for every positive integer $n$. The paper records that those authors show there are infinitely many $b$-Sierpiński numbers for every base $b>1$.

Statement

"Let $b$ be a positive integer such that $b$ is not a Mersenne number ($b+1$ is not a power of 2). There exist infinitely many $b$-Sierpiński numbers $k$ such that $k\cdot b^n+1$ has at least three distinct prime divisors for all positive integers $n$." (p. 1747)

The hypothesis excludes $b=1$ and every $b$ of the form $2^j-1$; the paper describes the result as known for $b=2$ and proves the case $b>2$.

Proof pointer. For $b>2$ the proof of Theorem 2 makes the Section 4 covering a $(b,1)$-primitive $3$-covering. Each modulus $m_i$ receives a primitive prime divisor $p_i$ of $b^{m_i}-1$, and the Chinese Remainder Theorem gives infinitely many $k$ with $k\cdot b^{r_i}+1\equiv0 \pmod{p_i}$ for every $i$, $k\equiv0\pmod{b-1}$ and $k\equiv1\pmod b$; the last two conditions give the coprimality and non-power clauses (pp. 1747--1748). The argument was followed but not independently checked here.