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

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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).

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