Inmathematics, aRiesel number is anoddnatural numberk for which${\displaystyle k\times 2^n-1}$ iscomposite for all natural numbersn (sequenceA101036 in theOEIS). In other words, whenk is a Riesel number, all members of the followingset are composite:

${\displaystyle \left\{k\times 2^n-1:n\in \mathbb{N}\right\}.}$

If the form is instead${\displaystyle k\times 2^n+1}$, thenk is aSierpiński number.

In 1956,Hans Riesel showed that there are aninfinite number of integersk such that${\displaystyle k\times 2^n-1}$ is notprime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positiveinteger multiple of 11184810.^[1] TheRiesel problem consists in determining the smallest Riesel number. Because nocovering set has been found for anyk less than 509203, it isconjectured to be the smallest Riesel number.

To check if there arek < 509203, theRiesel Sieve project (analogous toSeventeen or Bust forSierpiński numbers) started with 101 candidatesk. As of December 2022, 57 of thesek had been eliminated by Riesel Sieve,PrimeGrid, or outside persons.^[2] The remaining 41 values ofk that have yielded only composite numbers for all values ofn so far tested are

23669, 31859, 38473, 46663, 67117, 74699, 81041, 121889, 129007, 143047, 161669, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 485557, 494743.

The most recent elimination was in August 2024, when 107347 × 2^23427517 − 1 was found to be prime by Ryan Propper. This number is 7,052,391 digits long.

As of January 2023, PrimeGrid has searched the remaining candidates up ton = 14,900,000.^[3]

The sequence of currentlyknown Riesel numbers begins with:

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, ... (sequenceA101036 in theOEIS)

A number can be shown to be a Riesel number by exhibiting acovering set: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:

• ${\displaystyle 509203\times 2^n-1}$ has covering set {3, 5, 7, 13, 17, 241}
• ${\displaystyle 762701\times 2^n-1}$ has covering set {3, 5, 7, 13, 17, 241}
• ${\displaystyle 777149\times 2^n-1}$ has covering set {3, 5, 7, 13, 19, 37, 73}
• ${\displaystyle 790841\times 2^n-1}$ has covering set {3, 5, 7, 13, 19, 37, 73}
• ${\displaystyle 992077\times 2^n-1}$ has covering set {3, 5, 7, 13, 17, 241}.

Here is a sequence${\displaystyle a(k)}$ fork = 1, 2, .... It is defined as follows:${\displaystyle a(k)}$ is the smallestn ≥ 0 such that${\displaystyle k\cdot 2^n-1}$ is prime, or −1 if no such prime exists.

2, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 4, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 1, 0, 3, 1, 2, 0, 7, 0, 1, 3, 4, 0, 1, 2, 1, 1, 2, 0, 1, 2, 1, 3, 12, 0, 3, 0, 2, 1, 4, 1, 5, 0, 1, 1, 2, 0, 7, 0, 1, ... (sequenceA040081 in theOEIS). The first unknownn is for thatk = 23669.

Related sequences are (sequenceA050412 in theOEIS) (not allowingn = 0), for oddks, see (sequenceA046069 in theOEIS) or (sequenceA108129 in theOEIS) (not allowingn = 0).

A number both Riesel andSierpiński is aBrier number. The five smallest known examples (and note that some might be smaller, i.e. that the sequence might not be comprehensive) are: 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ...^[4](A076335).^[5]

Thedual Riesel numbers are defined as the odd natural numbersk such that |2ⁿ −k| is composite for all natural numbersn. There is a conjecture that the set of this numbers is the same as the set of Riesel numbers. For example, |2ⁿ − 509203| is composite for all natural numbersn, and 509203 is conjectured to be the smallest dual Riesel number.

The smallestn which 2ⁿ −k is prime are (for oddks, and this sequence requires that 2ⁿ >k)

2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, 10, 7, 7, 8, 7, 7, 7, 47, 8, 14, 9, 11, 10, 9, 10, 8, 9, 8, 8, ... (sequenceA096502 in theOEIS)

The oddks whichk − 2ⁿ are all composite for all 2ⁿ <k (thede Polignac numbers) are

1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, ... (sequenceA006285 in theOEIS)

The unknown values^{[clarification needed]} ofks are (for which 2ⁿ >k)

1871, 2293, 25229, 31511, 36971, 47107, 48959, 50171, 56351, 63431, 69427, 75989, 81253, 83381, 84491, ...

One can generalize the Riesel problem to an integer baseb ≥ 2. ARiesel number baseb is a positive integerk such thatgcd(k − 1,b − 1) = 1. (if gcd(k − 1,b − 1) > 1, then gcd(k − 1,b − 1) is a trivial factor ofk×bⁿ − 1 (Definition of trivial factors for the conjectures: Each and everyn-value has the same factor))^[6][7] For every integerb ≥ 2, there are infinitely many Riesel numbers baseb.

Example 1: All numbers congruent to 84687 mod 10124569 and not congruent to 1 mod 5 are Riesel numbers base 6, because of the covering set {7, 13, 31, 37, 97}. Besides, thesek are not trivial since gcd(k + 1, 6 − 1) = 1 for thesek. (The Riesel base 6 conjecture is not proven, it has 3 remainingk, namely 1597, 9582 and 57492)

Example 2: 6 is a Riesel number to all basesb congruent to 34 mod 35, because ifb is congruent to 34 mod 35, then 6×bⁿ − 1 is divisible by 5 for all evenn and divisible by 7 for all oddn. Besides, 6 is not a trivialk in these basesb since gcd(6 − 1,b − 1) = 1 for these basesb.

Example 3: All squaresk congruent to 12 mod 13 and not congruent to 1 mod 11 are Riesel numbers base 12, since for all suchk,k×12ⁿ − 1 has algebraic factors for all evenn and divisible by 13 for all oddn. Besides, thesek are not trivial since gcd(k + 1, 12 − 1) = 1 for thesek. (The Riesel base 12 conjecture is proven)

Example 4: Ifk is between a multiple of 5 and a multiple of 11, thenk×109ⁿ − 1 is divisible by either 5 or 11 for all positive integersn. The first few suchk are 21, 34, 76, 89, 131, 144, ... However, all thesek < 144 are also trivialk (i. e. gcd(k − 1, 109 − 1) is not 1). Thus, the smallest Riesel number base 109 is 144. (The Riesel base 109 conjecture is not proven, it has one remainingk, namely 84)

Example 5: Ifk is square, thenk×49ⁿ − 1 has algebraic factors for all positive integersn. The first few positive squares are 1, 4, 9, 16, 25, 36, ... However, all thesek < 36 are also trivialk (i. e. gcd(k − 1, 49 − 1) is not 1). Thus, the smallest Riesel number base 49 is 36. (The Riesel base 49 conjecture is proven)

We want to find and proof the smallest Riesel number baseb for every integerb ≥ 2. It is a conjecture that ifk is a Riesel number baseb, then at least one of the three conditions holds:

1. All numbers of the formk×bⁿ − 1 have a factor in some covering set. (For example,b = 22,k = 4461, then all numbers of the formk×bⁿ − 1 have a factor in the covering set: {5, 23, 97})
2. k×bⁿ − 1 has algebraic factors. (For example,b = 9,k = 4, thenk×bⁿ − 1 can be factored to (2×3ⁿ − 1) × (2×3ⁿ + 1))
3. For somen, numbers of the formk×bⁿ − 1 have a factor in some covering set; and for all othern,k×bⁿ − 1 has algebraic factors. (For example,b = 19,k = 144, then ifn is odd, thenk×bⁿ − 1 is divisible by 5, ifn is even, thenk×bⁿ − 1 can be factored to (12×19^n/2 − 1) × (12×19^n/2 + 1))

In the following list, we only consider those positive integersk such that gcd(k − 1,b − 1) = 1, and all integern must be ≥ 1.

Note:k-values that are a multiple ofb and wherek−1 is not prime are included in the conjectures (and included in the remainingk withred color if no primes are known for thesek-values) but excluded from testing (Thus, never be thek of "largest 5 primes found"), since suchk-values will have the same prime ask /b.

Conjectured smallest Riesel number basen are (start withn = 2)

509203, 63064644938, 9, 346802, 84687, 408034255082, 14, 4, 10176, 862, 25, 302, 4, 36370321851498, 9, 86, 246, 144, 8, 560, 4461, 476, 4, 36, 149, 8, 144, 4, 1369, 134718, 10, 16, 6, 287860, 4, 7772, 13, 4, 81, 8, 15137, 672, 4, 22564, 8177, 14, 3226, 36, 16, 64, 900, 5392, 4, 6852, 20, 144, 105788, 4, 121, 13484, 8, 187258666, 9, ... (sequenceA273987 in theOEIS)

• Sierpiński number
• Woodall number
• Experimental mathematics
• BOINC
• PrimeGrid

• Guy, Richard K. (2004).Unsolved Problems in Number Theory. Berlin:Springer-Verlag. p. 120.ISBN 0-387-20860-7.
• Ribenboim, Paulo (1996).The New Book of Prime Number Records. New York:Springer-Verlag. pp. 357–358.ISBN 0-387-94457-5.

• PrimeGrid
• The Riesel Problem: Definition and Status
• The Prime Glossary: Riesel number
• List of primes of the form: k*2^n-1, k<300
• List of primes of the form: k*2^n-1, k<300, Project Riesel Prime Search
• Riesel and Proth Prime Database

• Analytic number theory
• Unsolved problems in number theory
• Prime numbers

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