Problems & Puzzles:
Problems
Problem 58.
Brier numbers revisited
A Brier number is a number that is at
the same time Sierpinski and a Riesel number, that is to say is "an
integer k such that at the same time k.2n+1 and k.2n-1 are composite for
any value of n."
Perhaps is a good idea before start reading this Problem 58 to read the
Problem 29.
The story of the computation of the smallest Brier number can be
summarized in this Table:
Year |
Author |
Brier number |
Digits |
1975 |
F. Cohen & J.L.Selfridge |
47867742232066880047611079 |
26 |
2007 |
Filaseta et al |
143665583045350793098657 |
24 |
2013 |
Dan Ismailescu & P. S. Park |
10439679896374780276373 |
23 |
2014 |
Christope Clavier |
3316923598096294713661 |
22 |
From the other side Arkadiusz Wesolowski stated that "There are no Brier
numbers below 10^9.", See A076335.
Q. Can you compute a smaller Eric Brier number?
Contributions came from Emmanuel Vantieghem
***
Emmanuel wrote:
A couple of years ago I tried to find Brier
numbers. I found many, but they all were too big to be interesting.
The reason was that I thought that finding all coverings of the
natural numbers with twelve moduli was impossible.
Now, because you posed the question again, I tried to find them all.
To my great surprise, there are only 34560 solutions which I could
find in a couple of hours.
This allowed me to refind all Clavier's solutions plus six other
ones < 107711321583468432196343 :
Here you find :
the Brier numbers
the Sierpinski prime set
the Riesel prime set
11615103277955704975673
{3,
5, 17, 97, 241, 257, 673}
{3,
7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 331}
17855036657007596110949
{3,
7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
{3,
5, 13, 17, 97, 257, 673}
32904995562220857573541
{3,
5, 13, 17, 97, 257, 673}
{3,
7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
39842638540982216997209
{3,
7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
{3,
5, 13, 17, 97, 257, 673}
65548720000639644141973
{3,
7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
{3,
5, 13, 17, 97, 257, 673}
99970730871809953933447
{3,
7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
{3,
5, 13, 17, 97, 257, 673}
I think Clavier's record will stand for a very
long time (though I hope I live long enough to be wrong).
***
On august 7, 2023 Arkadiusz Wesolowski wrote:
At least three 23-digit
Brier numbers have been omitted.
38410495110832829780407
{3, 7, 11, 13, 19,
31, 37, 41, 61, 73, 151, 331, 1321}
{3, 5, 17, 97, 241,
257, 673}
56284389701328043058161
{3, 5, 17, 97, 241,
257, 673}
{3, 7, 11, 13, 19,
31, 37, 41, 61, 73, 109, 151}
64008563165679686026087
{3, 7, 11, 13, 19,
31, 37, 41, 61, 73, 109, 1321}
{3, 5, 17, 97, 241,
257, 673}
***
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