Problems & Puzzles:
Problems
Problem 69. More on
Brier numbersII.
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^n1
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^n1, 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.
***
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}.
***
