In mathematics, a Riesel number is an odd natural number k for which k × 2 n − 1 {\displaystyle k\times 2^{n}-1} is composite for all natural numbers n (sequence A101036 in the OEIS). In other words, when k is a Riesel number, all members of the following set are composite:
{ k × 2 n − 1 : n ∈ N } . {\displaystyle \left\{\,k\times 2^{n}-1:n\in \mathbb {N} \,\right\}.}
If the form is instead k × 2 n + 1 {\displaystyle k\times 2^{n}+1} , then k is a Sierpiński number.
Riesel problem
In 1956, Hans Riesel showed that there are an infinite number of integers k such that k × 2 n − 1 {\displaystyle k\times 2^{n}-1} is not prime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured to be the smallest Riesel number. To check if there are k < 509203, the Riesel Sieve project (analogous to Seventeen or Bust for Sierpiński numbers) started with 101 candidates k. As of December 2022, 57 of these k had been eliminated by Riesel Sieve, PrimeGrid, or outside persons. The remaining 41 values of k that have yielded only composite numbers for all values of n 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 × 223427517 − 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 to n = 14,900,000.
Known Riesel numbers The sequence of currently known 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, ... (sequence A101036 in the OEIS)
Covering set A number can be shown to be a Riesel number by exhibiting a covering 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:
509203 × 2 n − 1 {\displaystyle 509203\times 2^{n}-1} has covering set {3, 5, 7, 13, 17, 241}
762701 × 2 n − 1 {\displaystyle 762701\times 2^{n}-1} has covering set {3, 5, 7, 13, 17, 241}
777149 × 2 n − 1 {\displaystyle 777149\times 2^{n}-1} has covering set {3, 5, 7, 13, 19, 37, 73}
790841 × 2 n − 1 {\displaystyle 790841\times 2^{n}-1} has covering set {3, 5, 7, 13, 19, 37, 73}
992077 × 2 n − 1 {\displaystyle 992077\times 2^{n}-1} has covering set {3, 5, 7, 13, 17, 241}.
The smallest n for which k · 2n − 1 is prime Here is a sequence a ( k ) {\displaystyle a(k)} for k = 1, 2, .... It is defined as follows: a ( k ) {\displaystyle a(k)} is the smallest n ≥ 0 such that k ⋅ 2 n − 1 {\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, ... (sequence A040081 in the OEIS). The first unknown n is for that k = 23669. Related sequences are (sequence A050412 in the OEIS) (not allowing n = 0), for odd ks, see (sequence A046069 in the OEIS) or (sequence A108129 in the OEIS) (not allowing n = 0).
Simultaneously Riesel and Sierpiński A number both Riesel and Sierpiński is a Brier 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, ... (A076335).
The dual Riesel problem The dual Riesel numbers are defined as the odd natural numbers k such that |2n − k| is composite for all natural numbers n. 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 numbers n, and 509203 is conjectured to be the smallest dual Riesel number. The smallest n which 2n − k is prime are (for odd ks, and this sequence requires that 2n > k)
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