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Statement

Conjecture 7 (p. 11, quoted), which the paper attributes to Y.-G. Chen (J. Number Theory 98 (2003), 310--319). "For any positive integer $r$, there exist infinitely many positive odd numbers $k$ such that $k^r-2^n$ has at least two distinct prime factors for all positive integers $n$." The paper records that Chen settled it for $r$ odd and for $r$ twice an odd number with $3\nmid r$, that $r=4$ and $r=6$ are the least exponents his arguments do not reach, and that the full conjecture remains open (p. 11).

Theorem 15 (pp. 21--22, quoted). "There exist infinitely many positive odd numbers $k$ such that $k^4-2^n$ has at least two distinct prime factors for each positive integer $n$."

Corollary 21 (p. 28, quoted). "There exist infinitely many positive odd numbers $k$ such that $k^42^n-1$ has at least two distinct prime factors for each positive integer $n$." The paper notes that, by Lemma 4, the $k$ of Corollary 21 can be taken to be the same as those of Theorem 15 (p. 28).

Corollary 22 (p. 28). There is a set $\mathcal T$ of positive integers $r$ of positive asymptotic density such that (i) $4\mid r$ 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 obtains it from the $r=4$ covering for exponents $r=4m$ with $m$ coprime to $p-1$ for every prime $p$ of that covering (p. 28), and 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: Conjecture 7 on p. 11, Section 5 on pp. 21--31 with Theorem 15 on pp. 21--22, Lemmas 16 to 18 on pp. 22--23, Table 9 on pp. 24--26, Theorem 19 and Lemma 20 on p. 27, Corollaries 21 and 22 on p. 28.

Read depth. Claims checked: Conjecture 7, Theorem 15 and Corollaries 21 and 22 were read clause by clause on the page images. The proof was read but not checked step by step, and the 63-row covering of Table 9 was not recomputed. Nothing here is independently reviewed.

Proof pointer

Pp. 21--28. The proof works with $k^42^n-1$ and transfers to $k^4-2^n$ by Lemma 4 (p. 8) with $\mathcal S=\mathbb Z$. Lemma 16 (p. 22): if 2 is an $r$-th power modulo an odd prime $p$ and has order $m$ there, then for any class $a\pmod m$ some class for $k$ modulo $p$ makes $p$ divide $k^r2^n-1$ on that class of $n$; Lemma 17 (p. 22) tests whether 2 is an $r$-th power modulo $p$. Lemma 18 (p. 23) asserts that the 63 congruences of Table 9 (pp. 24--26) cover the integers, with distinct primes $p_i$, $\operatorname{ord}_{p_i}(2)=m_i$, and 2 a fourth power modulo $p_i$ for $i\ge2$; the paper describes how to check the covering modulo 997920 with reductions to 12960. The Chinese remainder theorem gives a progression of $k$ with a prime factor in $\mathcal P=\{p_1,\ldots,p_{63}\}$, and Lemma 20 (p. 27), proved from the theorem of Darmon and Granville on generalized Fermat equations (Theorem 19), gives a prime factor outside $\mathcal P$ for $k$ large. Corollary 21 uses Lemma 14 (p. 18) in place of Lemma 20.