Riesel Problem
In 1956, Hans Riesel showed that there are anKnown 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, ...Covering set
A number can be shown to be a Riesel number by exhibiting a ''The smallest ''n'' for which ''k'' · 2''n'' − 1 is prime
Here is a sequence for ''k'' = 1, 2, .... It is defined as follows: is the smallest ''n'' ≥ 0 such that 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, ... . The first unknown ''n'' is for that ''k'' = 23669. Related sequences are (not allowing ''n'' = 0), for odd ''k''s, see or (not allowing ''n'' = 0)Simultaneously Riesel and Sierpiński
A number may be simultaneously Riesel and Sierpiński. These are called Brier numbers. The five smallest known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... ().The dual Riesel problem
The dual Riesel numbers are defined as the odd natural numbers ''k'' such that , 2''n'' - ''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, , 2''n'' - 509203, is composite for all natural numbers ''n'', and 509203 is conjectured to be the smallest dual Riesel number. The smallest ''n'' which 2''n'' - ''k'' is prime are (for odd ''k''s, and this sequence requires that 2''n'' > ''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, ... The odd ''k''s which ''k'' - 2''n'' are all composite for all 2''n'' < ''k'' (the de 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, ... The unknown values of ''k''s are (for which 2''n'' > ''k'') :1871, 2293, 25229, 31511, 36971, 47107, 48959, 50171, 56351, 63431, 69427, 75989, 81253, 83381, 84491, ...Riesel number base ''b''
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. (if gcd(''k'' − 1, ''b'' − 1) > 1, then gcd(''k'' − 1, ''b'' − 1) is a trivial factor of ''k''×''b''''n'' − 1 (Definition of trivial factors for the conjectures: Each and every ''n''-value has the same factor)) For every integer ''b'' ≥ 2, there are infinitely many Riesel numbers base ''b''. 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 . Besides, these ''k'' are not trivial since gcd(''k'' + 1, 6 − 1) = 1 for these ''k''. (The Riesel base 6 conjecture is not proven, it has 3 remaining ''k'', namely 1597, 9582 and 57492) Example 2: 6 is a Riesel number to all bases ''b'' congruent to 34 mod 35, because if ''b'' is congruent to 34 mod 35, then 6×''b''''n'' − 1 is divisible by 5 for all even ''n'' and divisible by 7 for all odd ''n''. Besides, 6 is not a trivial ''k'' in these bases ''b'' since gcd(6 − 1, ''b'' − 1) = 1 for these bases ''b''. 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''×12''n'' − 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''×109''n'' − 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, ... However, all these ''k'' < 144 are also trivial ''k'' (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 remaining ''k'', namely 84) Example 5: If ''k'' is square, then ''k''×49''n'' − 1 has algebraic factors for all positive integers ''n''. The first few positive squares are 1, 4, 9, 16, 25, 36, ... However, all these ''k'' < 36 are also trivial ''k'' (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 base ''b'' for every integer ''b'' ≥ 2. It is a conjecture that if ''k'' is a Riesel number base ''b'', then at least one of the three conditions holds: # All numbers of the form ''k''×''b''''n'' − 1 have a factor in some covering set. (For example, ''b'' = 22, ''k'' = 4461, then all numbers of the form ''k''×''b''''n'' − 1 have a factor in the covering set: ) # ''k''×''b''''n'' − 1 has algebraic factors. (For example, ''b'' = 9, ''k'' = 4, then ''k''×''b''''n'' − 1 can be factored to (2×3''n'' − 1) × (2×3''n'' + 1)) # For some ''n'', numbers of the form ''k''×''b''''n'' − 1 have a factor in some covering set; and for all other ''n'', ''k''×''b''''n'' − 1 has algebraic factors. (For example, ''b'' = 19, ''k'' = 144, then if ''n'' is odd, then ''k''×''b''''n'' − 1 is divisible by 5, if ''n'' is even, then ''k''×''b''''n'' − 1 can be factored to (12×19''n''/2 − 1) × (12×19''n''/2 + 1)) In the following list, we only consider those positive integers ''k'' such that gcd(''k'' − 1, ''b'' − 1) = 1, and all integer ''n'' must be ≥ 1. Note: ''k''-values that are a multiple of ''b'' and where ''k''−1 is not prime are included in the conjectures (and included in the remaining ''k'' with color if no primes are known for these ''k''-values) but excluded from testing (Thus, never be the ''k'' of "largest 5 primes found"), since such ''k''-values will have the same prime as ''k'' / ''b''. Conjectured smallest Riesel number base ''n'' are (start with ''n'' = 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, ...See also
* Sierpiński number *References
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