---

Statement

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

Theorem 5 (p. 9, quoted). "A positive proportion of the positive integers are simultaneously Sierpiński and Riesel numbers. The number 143665583045350793098657 is one such number."

The paper states that the first sentence is not new and follows from unpublished work of E. Brier (1998), whose example had 41 digits; Y. Gallot (2000, unpublished) found one with 27 digits (p. 9). The new content is the 24-digit example.

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 5 on p. 9, its proof with Tables 3 and 4 on pp. 9--11.

Read depth. Claims checked: the statement was read clause by clause on the page images. The proof was read; in addition, a direct computation for this page confirmed that the 24-digit number is odd and lies in every residue class for $k$ listed in Tables 3 and 4, that the classes for $n$ in each table cover the integers, and that each row's prime divides $k\cdot2^n+1$ (Table 3) or $k\cdot2^n-1$ (Table 4) on its class. Nothing here is independently reviewed.

Proof pointer

Pp. 9--11. Table 3 (p. 10) lists twelve pairs of a class $a\pmod m$ for $n$ and a class $b\pmod p$ for $k$, with $\operatorname{ord}_p(2)=m$ and $b2^a+1\equiv0\pmod p$, using the primes 3, 7, 73, 19, 37, 109, 31, 11, 151, 331, 61 and 1321; the classes for $n$ cover the integers, checked modulo their least common multiple 180, so for every $k$ in all the $k$-classes and every positive $n$, one of these primes divides $k\cdot2^n+1$. Table 4 does the same for $k\cdot2^n-1$ with seven pairs, the primes 3, 5, 17, 257, 13, 241 and 97, and least common multiple 48. The two tables share only the prime 3, where both ask for $k\equiv1\pmod3$; adding $k\equiv1\pmod2$, the Chinese remainder theorem gives a full residue class of such $k$, hence a positive proportion, and the stated number is one of them (p. 11).