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Statement

Conjecture 7 (p. 11), attributed to Y.-G. Chen, asks for every positive integer $r$ for infinitely many positive odd $k$ such that $k^r-2^n$ has at least two distinct prime factors for all positive integers $n$; it is quoted, with what Chen settled, on the page of Theorem 15, the case $r=4$.

Theorem 23 (p. 28, quoted). "There exist infinitely many positive odd numbers $k$ such that $k^6-2^n$ has at least two distinct prime factors for each positive integer $n$."

Corollary 25 (p. 29, quoted). "There exist infinitely many positive odd numbers $k$ such that $k^62^n-1$ has at least two distinct prime factors for each positive integer $n$."

Corollary 26 (p. 29). There is a set $\mathcal T'$ of positive integers $r$ of positive asymptotic density such that (i) $6\mid r$ for every $r\in\mathcal T'$, and (ii) for each $r\in\mathcal T'$ there are infinitely many positive odd $k$ such that each of $k^r-2^n$ and $k^r2^n-1$ has at least two distinct prime factors for each positive integer $n$. The paper states that the $r$ in $\mathcal T'$ are not covered by Chen's work (p. 29).

Source. M. Filaseta, C. Finch and M. Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, J. Number Theory 128 (2008), no. 7, 1916--1940, doi:10.1016/j.jnt.2008.02.004, read in the authors' preprint identified on the source card, whose pages are numbered 1 to 32 and carry no journal pagination: Theorem 23 on p. 28, Lemma 24 and Corollaries 25 and 26 on p. 29, Table 10 on pp. 30--31.

Read depth. Claims checked: the three statements were read clause by clause on the page images. The proof was read but not checked step by step, and the 49-row covering of Table 10 was not recomputed. Nothing here is independently reviewed.

Proof pointer

Pp. 28--29. The proof follows that of Theorem 15 with $r=6$ in Lemmas 16 and 17. The classes $n\equiv0\pmod2$ and $n\equiv0\pmod3$ are handled by 3 and 7 with $k\equiv1\pmod3$ and $k\equiv1\pmod7$; the remaining primes are chosen so that 2 is a sixth power modulo each. Lemma 24 (p. 29) asserts that the 49 congruences of Table 10 (pp. 30--31) cover the integers, with distinct primes $p_i$, $\operatorname{ord}_{p_i}(2)=m_i$, and 2 a sixth power modulo $p_i$ for $i\ge3$; the covering is checked modulo 16800. The paper states that Lemma 24 and Theorem 23 then follow, and the corollaries as in the case $r=4$, where Lemma 4 transfers from $k^42^n-1$ to $k^4-2^n$ and Lemma 20 (or Lemma 14) supplies a prime factor outside the finite set.