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Updated Oct 7

Problem 1113

Statement. A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every $2^km+1$ is divisible by some $p\in P$ .

Are there Sierpinski numbers with no finite covering set of primes?

Status. Open.

Source. erdosproblems.com/1113, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1113, https://www.erdosproblems.com/1113.

References.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[FFK08] Filaseta, Michael and Finch, Carrie and Kozek, Mark, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture. J. Number Theory 128 (2008), no. 7, 1916-1940.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B21 " $k\cdot2^n+1$ composite for all $n$ ", pp. 119--121, gives Sierpiński's covering-congruence construction, Selfridge's $78557$ and Stanton's covering sets for $k\cdot2^n-1$ , and does not state the question; the site's commentary locates its precise formulation in Guy's problem F13. Section F13 "Covering systems of congruences" (from p. 382) records on p. 384 Erdős's conjecture that every sequence $d\cdot2^k+1$ ( $k=1,2,\dots$ ), $d$ fixed and odd, that contains no primes can be obtained from covering congruences, equivalently that the least prime factors of its terms are bounded. Library home: guy_2004_unsolved_problems_number_theory.
[Iz95] Izotov, Anatoly S., A note on Sierpiński numbers. Fibonacci Quart. (1995), 206-207.
[Si60] Sierpiński, W., Sur un problème concernant les nombres $k\cdot 2\sp{n}+1$. Elem. Math. (1960), 73-74.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (35)

Covering Systems Covering Systems Baillie et al.: The problem of Sierpiński concerning 𝑘⋅2ⁿ+1 Covering sets recalled (p. 229): k = 201446503145165177, 271129 and 78557 Main result (pp. 229, 231): the least Sierpiński number is one of 119 values from 3061 to 78557 Table 1 (p. 230): the eight odd k below 10000 with no known prime k 2^n + 1 Table 2 (p. 230): least exponents n >= 3000 giving primes k 2^n + 1 Table 3 (p. 230): the odd k between 10000 and 78557 with k 2^n + 1 composite for all n <= 2000 Banks et al.: Sierpiński and Carmichael numbers Corollary 1 (p. 356): a positive-lower-density set of k with every 2^n k + 1 neither prime nor Carmichael Proposition 1 (p. 369): at least a constant times x^(1/5) Sierpiński Carmichael numbers up to x Theorem 1 (p. 356): almost every odd k has no Carmichael number of the form 2^n k + 1 Theorem 2 (p. 356): infinitely many numbers are Sierpiński, Riesel and Carmichael at once Chen: On integers of the form 𝑘2ⁿ+1 Corollary (p. 356): positive lower density of odd k with every k 2^n + 1 having at least three distinct prime factors Lemma 2 (p. 357): few odd M have some M 2^n + 1 composed of r primes from a fixed finite set Theorem 1 (p. 356): a (2,1)-primitive r-covering gives positive lower density for r+1 prime factors Theorem 2 (p. 356): (2,1)-primitive r-coverings are equivalent to finite prime sets giving r divisors of every k 2^n + 1 Chen: On integers of the forms kʳ−2ⁿ and kʳ2ⁿ+1 Corollary 1 (p. 311): for odd r, infinitely many primes p make every p^r - 2^n, or every p^r 2^n + 1, have two distinct prime factors Corollary 2 (p. 312): for odd r with 3 not dividing r, infinitely many primes p make every p^(2r) - 2^n, or every p^(2r) 2^n + 1, have two distinct prime factors Theorem 1 (p. 311): for odd r, the odd k with every k^r - 2^n, or every k^r 2^n + 1, having two distinct prime factors contain an infinite progression Theorem 2 (p. 311): for odd r with 3 not dividing r, the odd k with every k^(2r) - 2^n, or every k^(2r) 2^n + 1, having two distinct prime factors contain an infinite progression Erdös–Odlyzko: On the density of odd integers of the form (p − 1)2−n and related questions Conjecture and question on p. 258: the density of k 2^n + 1 and covering congruences Theorem 1: odd k with a prime k 2^n + 1 have positive lower density Theorem 2: positive lower density for several prime bases covering_systems/filaseta_2008_powers_associated_sierpinski_numbers_riesel Theorem 10: l^4 is a Sierpinski number for l = 44745755 modulo 2*3*5*17*97*241*257*673 covering_systems/izotov_1995_note_sierpinski_numbers Theorem 1: fourth powers that are Sierpinski numbers SUN–FANG: ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2^−n IN ARITHMETIC PROGRESSIONS Corollary (p. 433, unnumbered): covering systems and density zero Theorem (p. 433, unnumbered): the coprimality dichotomy for progressions of odd k number_theory/guy_2004_unsolved_problems_number_theory

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