Problems & Puzzles: Problems

Problem 69. More on Brier numbers-II.

Brier numbers have been discussed in our Problems 29, 49, 52, 58 & 68.

Let's remember the basics:

• A Brier number k, is such that k*2^n+1 & k*2^n-1 are composite for all n integer values.
• Moreover, there are separated covering sets of primes A & B for each k*2^n+1 & k*2^n-1, respectively.

Example by Cohen & Selfridge (1974):

For k=47867742232066880047611079
A={3, 5, 13, 17, 97, 257, 241}
B={3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}

In this example the sets of primes A & B share just one prime number, "3"

As far as I know, it has not been reported any Brier number such that A & B are disjoint sets (no one single prime integer is shared)

Q. Can you find one Brier number such that A & B are disjoint sets, or prove that these are impossible?

Emmanuel Vantieghem wrote on Set 3, 2017:

I could not find a Brier number without  3  in the covering prime set.
This should be a number that is divisible by 3.

However, it was allready fairly difficult to find a Sierpinski number that was a multiple of 3.
Here is the smallest I found :
145530372006115980757765448958528778528869413629687401
The covering prime set is :
{5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 71, 73, 97, 109, 113, 127, 151, 257, 331, 337, 631, 23311, 61681, 122921}.

The smallest Riesel number that is a multiple of 3 I could find is :
70427434562856047803813273709635465686724448675347209
with the same covering prime set.

Since I used many 'small' prime numbers to get a covering, I think it is highly difficult to find a second covering that would lead to a Brier number.

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Arkadiusz Wesolowski wrote on Set 4, 2017:

Note that if k is a Sierpinski (or Riesel) number not divisible by 3, and k has the covering set S = {p(1), p(2), ..., p(s)} with p(1) > 3, then N = k*2^n + 1 (N = k*2^n - 1) is composite for all n >= 1, and every n is covered by at least one of six congruences, where 3 | N <==> n == 0 (mod 2) or 3 | N <==> n == 1 (mod 2). So k has at least two covering sets.

Example:
k = 222252191206502278417217 is a Sierpinski number, k == 2 (mod 3).
k has the covering set {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331} as well as the covering set {3, 5, 7, 13, 17, 241}.

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