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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 $n$.

Conjecture 2 (p. 5, quoted), which the paper attributes to Erdős through Guy's Unsolved Problems in Number Theory (3rd ed., 2004), Section F13. "If $k$ is a Sierpiński number, then the smallest prime divisor of $k\cdot2^n+1$ is bounded as $n$ tends to infinity." The introduction (p. 2) calls this conjecture "Conjecture 1 in the next section"; the statement in Section 2 is numbered Conjecture 2. The paper presents it as the precise form of Erdős's belief that every Sierpinski number is obtainable from an argument involving a covering (p. 2).

Theorem 10 (p. 14, quoted). "If $\ell$ is a positive integer satisfying

$$\ell\equiv44745755\pmod{2\cdot3\cdot5\cdot17\cdot97\cdot241\cdot257\cdot673},$$

then $k=\ell^4$ is a Sierpiński number."

The paper compares 44745755 with the 15-digit $\ell=734110615000775$ that Izotov's own construction gives at its least (p. 6). For $n\not\equiv2\pmod4$ every $k\cdot2^n+1$ has a prime factor in $\{3,17,97,241,257,673\}$; for $n\equiv2\pmod4$ the term is composite through the factorization (1) (p. 6)

$$\ell^4\cdot2^{4u+2}+1=4(\ell\cdot2^u)^4+1 =\bigl(\ell^2 2^{2u+1}+\ell\,2^{u+1}+1\bigr)\bigl(\ell^2 2^{2u+1}-\ell\,2^{u+1}+1\bigr),$$

and the paper imposes $\ell\equiv0\pmod5$ to ensure that the smallest prime divisor of $k\cdot2^n+1$ is not always taken from $\{3,17,97,241,257,673\}\cup\{5\}$ (p. 14).

Evidence against Conjecture 2 (pp. 6--7 and 14--15, not a theorem). The paper states that it cannot conclude that $44745755^4$ does not arise from a covering argument (p. 14), and calls a proof that any of its examples cannot arise from a covering "out of reach" (p. 2). It offers Table 5 (p. 15), the smallest prime factors of $k\cdot2^n+1$ for $k=44745755^4$ at $n=54$, $90$ and $214$ (5719237, 64450569241 and 338100368290543455397, with the factorizations of the order of 2 modulo each), and Table 2 (p. 7) for Izotov's number, as evidence that these $k$ are counterexamples to Conjecture 2.

Conjecture 3 (p. 7, quoted). "If $k$ is a Sierpiński number that is not of the form $\ell^r$ for some integers $\ell\ge1$ and $r>1$, then the smallest prime divisor of $k\cdot2^n+1$ is bounded as $n$ tends to infinity."

Open questions (p. 3). In connection with Conjecture 2 the paper asks whether there is a method to determine whether the smallest prime divisor of $k\cdot2^n+1$ is bounded for a given $k$, whether one can prove that the smallest prime divisor of $5\cdot2^n+1$ is not bounded as $n$ tends to infinity, and the same for $11\cdot2^n-1$. It notes that the question for $5$ has a positive answer when $5$ is replaced by a smaller positive integer, and shows it for $3$: for any $x$ some $n$ has every odd prime $\le x$ dividing $2^n-1$, so the smallest prime factor of $3\cdot2^n+1$ exceeds $x$.

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: the open questions on p. 3, Conjecture 2 on p. 5, Izotov's construction and the factorization (1) on p. 6, Table 2 and Conjecture 3 on p. 7, Section 3 on pp. 13--17 with Theorem 10 on p. 14 and Table 5 on p. 15.

Read depth. Claims checked: Theorem 10, Conjectures 2 and 3, and the open questions were read clause by clause on the page images. A direct computation for this page confirmed that 44745755 is odd and divisible by 5, satisfies the six congruences on $\ell$ of p. 14, that each row's prime divides $\ell^4\cdot2^n+1$ on its class for $n$, and that the six classes for $n$ together with $n\equiv2\pmod4$ cover the integers modulo 48. The tables of smallest prime factors were not recomputed. Nothing here is independently reviewed.

Proof pointer

Pp. 13--14. The six implications printed on p. 14 pair the classes $n\equiv1\pmod2$, $4\pmod8$, $32\pmod{48}$, $0\pmod{24}$, $8\pmod{16}$ and $16\pmod{48}$ with $\ell\equiv2\pmod3$, $4\pmod{17}$, $43\pmod{97}$, $8\pmod{241}$, $256\pmod{257}$ and $4\pmod{673}$; each is justified by $\operatorname{ord}_p(2)=m$ and $b^4 2^a+1\equiv0\pmod p$. (In the last implication the modulus of the conclusion is printed as 637; the congruence on $\ell$ and the set $\mathcal P$ show that 673 is meant.) With $n\equiv2\pmod4$, handled by the factorization (1), these classes cover the integers, and adding $\ell\equiv1\pmod2$ and $\ell\equiv0\pmod5$ gives the residue class of Theorem 10.

Bears on

• Problem 1113: the problem asks whether some Sierpinski number has no finite covering set of primes. A finite covering set exists exactly when the smallest prime divisor of $k\cdot2^n+1$ stays bounded (an observation of this page), so the problem asks whether Conjecture 2 fails. Theorem 10 supplies an infinite family of Sierpinski numbers $\ell^4$ that the paper calls likely not obtainable by covering arguments (p. 2), and Table 5 gives computational evidence that its least member $44745755^4$ has no finite covering set, but the paper proves no such $k$ lacks one; the theorem therefore does not answer the problem. The problem counts $2^km+1$ from $k\ge0$ while the paper's definition starts at $n=1$; for odd $m>1$ the extra term $m+1$ is even and composite, so the two definitions agree.