In other words, when k is a Riesel number, all members of the following set are composite: If the form is instead
In 1956, Hans Riesel showed that there are an infinite number of integers k such that
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.
[2] The remaining 42 values of k that have yielded only composite numbers for all values of n so far tested are The most recent elimination was in April 2023, when 97139 × 218397548 − 1 was found to be prime by Ryan Propper.
The only proven Riesel numbers below one million have covering sets as follows: Here is a sequence
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, ...
The smallest n which 2n - k is prime are (for odd ks, and this sequence requires that 2n > k) The odd ks which k - 2n are all composite for all 2n < k (the de Polignac numbers) are The unknown values[clarification needed] of ks are (for which 2n > k) One can generalize the Riesel problem to an integer base b ≥ 2.
A Riesel number base b is a positive integer k such that gcd(k − 1, b − 1) = 1.
Example 3: All squares k congruent to 12 mod 13 and not congruent to 1 mod 11 are Riesel numbers base 12, since for all such k, k×12n − 1 has algebraic factors for all even n and divisible by 13 for all odd n. Besides, these k are not trivial since gcd(k + 1, 12 − 1) = 1 for these k. (The Riesel base 12 conjecture is proven) Example 4: If k is between a multiple of 5 and a multiple of 11, then k×109n − 1 is divisible by either 5 or 11 for all positive integers n. The first few such k are 21, 34, 76, 89, 131, 144, ...
(The Riesel base 109 conjecture is not proven, it has one remaining k, namely 84) Example 5: If k is square, then k×49n − 1 has algebraic factors for all positive integers n. The first few positive squares are 1, 4, 9, 16, 25, 36, ...
676373272×31072675−11068687512×31067484−11483575692×31067339−1780548926×31064065−11776322388×31053069−1 136804×54777253-1 52922×54399812-1 177742×54386703-1 213988×54138363-1 Conjectured smallest Riesel number base n are (start with n = 2)