

A076335


Brier numbers: numbers that are both Riesel and Sierpiński [Sierpinski], or odd n such that for all k >= 1 the numbers n*2^k + 1 and n*2^k  1 are composite.


27



3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, 21444598169181578466233, 28960674973436106391349, 32099522445515872473461, 32904995562220857573541
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OFFSET

1,1


COMMENTS

a(1), a(4), and a(6)a(8) computed by Christophe Clavier, Dec 31 2013 (see link below). 10439679896374780276373 had been found earlier in 2013 by Dan Ismailescu and Peter Seho Park (see reference below). a(3), a(5), and a(9) computed in 2014 by Emmanuel Vantieghem.
These are just the smallest examples known  there may be smaller ones.
Other Brier numbers are 143665583045350793098657, 1547374756499590486317191, 3127894363368981760543181, 3780564951798029783879299, but these may not be the /next/ Brier numbers after those shown. From 2002 to 2013 these four numbers were given here as the smallest known Brier numbers, so the new entry A234594 has been created to preserve that fact.  N. J. A. Sloane, Jan 03 2014
143665583045350793098657 computed in 2007 by Michael Filaseta, Carrie Finch, and Mark Kozek.
It is a conjecture that every such number has more than 10 digits. In 2011 I have calculated that for any n < 10^10 there is a k such that either n*2^k + 1 or n*2^k  1 has all its prime factors greater than 1321.  Arkadiusz Wesolowski, Feb 03 2016 [Editor's note: The comment below states that the conjecture is now proved.  M. F. Hasler, Oct 06 2021]
There are no Brier numbers below 10^10. For each n < 10^10, there exists at least one prime of the form n*2^k1 or n*2^k+1 with k <= 356981. The largest necessary prime is 1355477231*2^356981+1.  Kellen Shenton, Oct 25 2020


LINKS



CROSSREFS

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639, A076336, A076337, A040081, A040076, A103963, A103964, A038699, A050921, A064699, A052333, A003261, A364412, A364413.
A234594 is the old, incorrect version.


KEYWORD

nonn


AUTHOR



EXTENSIONS

Many terms reported in Problem 29 from "The Prime Problems & Puzzles Connection" from Carlos Rivera, May 30 2010


STATUS

approved



