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Statement

Setting (p. 1). A Riesel number is a positive odd integer $k$ such that $k\cdot2^n-1$ is composite for all positive integers $n$.

Theorem 11 (p. 16). "There are infinitely many squares that are Riesel numbers." Its second sentence gives the example

$$3896845303873881175159314620808887046066972469809^2.$$

The paper states that the first sentence is not new and is a consequence of Y.-G. Chen's work (J. Number Theory 98 (2003), 310--319); the new content is the example, which it presents as a square that appears not to arise from a covering argument and as the least it found by its method (pp. 16--17).

Polignac form (p. 17, unlabeled). For $k=\ell^2$ the Riesel number of Theorem 11, the paper deduces from Lemma 4 that $\lvert k-2^n\rvert$ is composite for all positive integers $n$. Lemma 4 (p. 8) states, for $\mathcal S$ the integers, the even integers or the odd integers, a finite set $\mathcal P$ of odd primes and an integer $k$: if for every sufficiently large $n\in\mathcal S$ some $p\in\mathcal P$ divides $k\cdot2^n-1$, then for each $n\in\mathcal S$ some $p\in\mathcal P$ divides $k-2^n$, and conversely with the two forms exchanged.

Evidence that the example avoids coverings (pp. 16--17, not a theorem). Tables 7 and 8 list smallest prime factors of $\ell^2 2^n-1$ and of $\lvert\ell^2-2^n\rvert$ with the orders of 2 modulo them; the paper adds that a sieve by the first 20000 primes found, for $n\le25000$, at least 170 different values of the least prime factor of $\ell^2 2^n-1$ and 59 values of $n$ with no prime factor among those primes.

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: Lemma 4 on p. 8, the construction on pp. 15--16, Table 6 and Theorem 11 on p. 16, Tables 7 and 8 and the Polignac form on p. 17.

Read depth. Claims checked: the statement, Lemma 4 and the Polignac deduction were read clause by clause on the page images. A direct computation for this page confirmed that the printed square root is odd and lies in each class for $\ell$ of Table 6, that the classes for $n$ together with $n\equiv0\pmod2$ cover the integers modulo 6720, and that each row's prime divides $\ell^2 2^n-1$ on its class for $n$. The tables of smallest prime factors were not recomputed. Nothing here is independently reviewed.

Proof pointer

Pp. 15--16. For $n=2u$ and $k=\ell^2$ with $\ell>1$, $k\cdot2^n-1=(\ell\cdot2^u+1)(\ell\cdot2^u-1)$ is composite, so only odd $n$ need a covering prime. Table 6 (p. 16) gives twenty pairs of a class for $n$ and a class for $\ell$, using the primes 7, 17, 31, 41, 71, 97, 113, 127, 151, 241, 257, 281, 337, 641, 673, 1321, 14449, 29191, 65537 and 6700417; with $n\equiv0\pmod2$ they cover the integers (least common multiple 6720), and every odd $\ell$ in all the classes makes $\ell^2$ a Riesel number. For the Polignac form, $\ell^2-2^{2u}=(\ell+2^u)(\ell-2^u)$ with $\lvert\ell-2^u\rvert>1$ since neither $\ell+1$ nor $\ell-1$ is a power of 2, and Lemma 4 with $\mathcal S$ the odd integers handles odd $n$, after checking that $k\pm p$ is not a power of 2 for $p\in\mathcal P$ (p. 17).