The Riesel problem

Show that 509203 is the smallest odd k with k·2^n − 1 composite for all n, by finding primes for the 41 remaining smaller k.

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1	# The Riesel problem
2
3	In 1956 Hans Riesel showed that $509203 \cdot 2^n - 1$ is composite for every $n$, because one of six small primes always divides it. Is 509203 the smallest odd number with this property? Settling the question means finding a prime of the form $k \cdot 2^n - 1$ for each of the 41 smaller values of $k$ still unresolved, and a distributed search is testing exponents beyond seventeen million.
4
5	## Problem
6
7	An odd positive integer $k$ is a Riesel number if $k \cdot 2^n - 1$ is composite for every integer $n \ge 1$.
8
9	Conjecture. The smallest Riesel number is $509203$. Equivalently, for every odd $k < 509203$ there is some $n \ge 1$ such that $k \cdot 2^n - 1$ is prime.
10
11	## Background
12
13	Riesel (1956) proved that $509203$ is a Riesel number: every number $509203 \cdot 2^n - 1$ is divisible by one of the primes in the covering set $\{3, 5, 7, 13, 17, 241\}$, which works because the residues of $2^n$ modulo these primes repeat with a common period. The question mirrors the Sierpiński problem for numbers $k \cdot 2^n + 1$. Most known Riesel numbers come from covering sets, although Filaseta, Finch and Kozek (2008) gave one that is believed not to. The remaining candidates below 509203 are being tested by the volunteer computing project PrimeGrid.
14
15	## What is known
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17	- $509203$ is a Riesel number (Riesel, 1956), and no smaller $k$ is known to have a covering set.
18	- PrimeGrid's statistics page, generated on 20 October 2026, lists 41 values of $k$ below $509203$ for which no prime $k \cdot 2^n - 1$ is known; the smallest is $k = 23669$. Every remaining candidate has been tested up to $n$ of about $1.71 \times 10^7$.
19	- Primes found during the search are very large; for example the table includes $107347 \cdot 2^{23427517} - 1$, which eliminated $k = 107347$.
20
21	## What would count as progress
22
23	- A prime $k \cdot 2^n - 1$ for any of the 41 remaining values of $k$. A claimed prime can be confirmed by an independent primality test, and each one reduces the problem.
24	- A proof, for some remaining $k$, that $k \cdot 2^n - 1$ is always composite, which would disprove the conjecture; this would most likely require a covering set or an algebraic factorisation.
25	- Faster sieving and primality-testing software for numbers of this form, measured by the throughput of the search.
26	- A formal proof of the Lean statement `conjecture` in [OEIS/108129.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OEIS/108129.lean), which encodes the problem through the OEIS sequence A108129.
27
28	## References
29
30	1. M. Filaseta, C. Finch and M. Kozek, [On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture](https://doi.org/10.1016/j.jnt.2008.02.004), J. Number Theory 128 (2008), 1916–1940.
31	2. OEIS Foundation, [A101036: Riesel numbers that have a covering set](https://oeis.org/A101036) and [A108129: the Riesel problem](https://oeis.org/A108129).
32	3. PrimeGrid, [The Riesel Problem statistics](https://www.primegrid.com/stats_trp_llr.php), retrieved 20 October 2026.
33
34	## Source
35
36	Imported from [Formal Conjectures OEIS/108129](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OEIS/108129.lean), curated by Google DeepMind and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.