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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 $n$. $F_m=2^{2^m}+1$ is the $m$-th Fermat number.

Theorem 8 (p. 12, quoted). "Suppose there exist at least $r$ composite Fermat numbers $F_m=2^{2^m}+1$. Then there are infinitely many positive odd integers $k$ such that if $t$ is a positive integer not divisible by $2^r$, then $k^t$ is a Sierpiński number."

Corollary 9 (p. 13, quoted). "There is a $k$ such that all of the numbers $k,k^2,k^3,\ldots,k^{3.45\cdot10^{69}}$ are simultaneously Sierpiński numbers."

The paper derives Corollary 9 from Theorem 8 with the count, taken from the web page of composite Fermat numbers at the time of writing, of 231 known $m$ with $F_m$ composite: Theorem 8 with $r=231$ gives a $k$ with $k^t$ Sierpinski for every $t<2^{231}$ (p. 13), and $2^{231}$ is about $3.45\cdot10^{69}$. The corollary therefore rests on that computational record of composite Fermat numbers, which the paper cites and does not prove. Theorem 1 of the paper removes the dependence on Fermat numbers for any fixed range of exponents.

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 8 and its proof on pp. 12--13, Corollary 9 on p. 13.

Read depth. Claims checked: both statements were read clause by clause on the page images. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 12--13. No Fermat number is a perfect power (by Bang's theorem on primitive prime divisors), so each composite $F_{m_j}$, $0\le j\le r-1$, has two distinct prime factors $p_j$ and $q_j$. The proof imposes $k\equiv1$ modulo 2, modulo the other $F_m$ with $m<m_{r-1}$ and modulo each $p_j$, and $k\equiv2^{2^{m_j-j}}\pmod{q_j}$. For $t=2^wt'$ with $w\le r-1$ and $t'$ odd, and $n=2^in'$ with $n'$ odd, the term $k^t2^n+1$ is divisible by $F_i$, by $p_j$ or by $q_w$ according as $i<m_w$ (with $i$ not among the $m_j$), $i=m_j$ for some $j\le w$, or $i>m_w$; every such $k$ other than possibly the least exceeds the product of these divisors.

Theorem 8 is the precursor of Theorem 1.