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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.

%I A076335 #135 Feb 16 2025 08:32:47
%S A076335 3316923598096294713661,10439679896374780276373,
%T A076335 11615103277955704975673,12607110588854501953787,
%U A076335 17855036657007596110949,21444598169181578466233,28960674973436106391349,32099522445515872473461,32904995562220857573541
%N 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.
%C A076335 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_.
%C A076335 These are just the smallest examples known - there may be smaller ones.
%C A076335 There are no Brier numbers below 10^9. - _Arkadiusz Wesolowski_, Aug 03 2009
%C A076335 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
%C A076335 143665583045350793098657 computed in 2007 by Michael Filaseta, Carrie Finch, and Mark Kozek.
%C A076335 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]
%C A076335 There are no Brier numbers below 10^10.  For each n < 10^10, there exists at least one prime of the form n*2^k-1 or n*2^k+1 with k <= 356981. The largest necessary prime is 1355477231*2^356981+1. - _Kellen Shenton_, Oct 25 2020
%H A076335 D. Baczkowski, J. Eitner, C. E. Finch, B. Suminski, and M. Kozek, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.html">Polygonal, Sierpinski, and Riesel numbers</a>, Journal of Integer Sequences, 2015 Vol 18. #15.8.1.
%H A076335 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/RieselNumber.html">Riesel number</a>
%H A076335 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/SierpinskiNumber.html">Sierpinski number</a>
%H A076335 Christophe Clavier, <a href="/A076335/a076335.txt">14 new Brier numbers</a>
%H A076335 Fred Cohen and J. L. Selfridge, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0376583-0">Not every number is the sum or difference of two prime powers</a>, Math. Comput. 29 (1975), pp. 79-81.
%H A076335 P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1950-07.pdf">On integers of the form 2^k + p and some related problems</a>, Summa Brasil. Math. 2 (1950), pp. 113-123.
%H A076335 M. Filaseta et al., <a href="http://www.math.sc.edu/~filaseta/papers/SierpinskiEtCoPapNew.pdf">On Powers Associated with Sierpiński Numbers, Riesel Numbers and Polignac’s Conjecture</a>, Journal of Number Theory, Volume 128, Issue 7, July 2008, Pages 1916-1940. (See pages 9-10)
%H A076335 Michael Filaseta and Jacob Juillerat, <a href="https://arxiv.org/abs/2101.08898">Consecutive primes which are widely digitally delicate</a>, arXiv:2101.08898 [math.NT], 2021.
%H A076335 Michael Filaseta, Jacob Juillerat, and Thomas Luckner, <a href="https://arxiv.org/abs/2209.10646">Consecutive primes which are widely digitally delicate and Brier numbers</a>, arXiv:2209.10646 [math.NT], 2022. See also <a href="http://math.colgate.edu/~integers/x75/x75.pdf">Integers</a> (2023) Vol. 23, #A75.
%H A076335 Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf">A search for some small Brier numbers</a>, 2000.
%H A076335 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/23122.html">Prime Curios! 6992565235279559197457863</a>
%H A076335 Dan Ismailescu and Peter Seho Park, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.html">On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences</a>, Journal of Integer Sequences, 16 (2013), #13.9.8.
%H A076335 Joe McLean, <a href="http://oeis.org/A076336/a076336b.html">Brier Numbers</a> [Cached copy]
%H A076335 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_029.htm">Problem 29. Brier numbers</a>, The Prime Puzzles and Problems Connection.
%H A076335 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_058.htm">Problem 58. Brier numbers revisited</a>, The Prime Puzzles and Problems Connection.
%H A076335 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_068.htm">Problem 68. More on Brier numbers</a>, The Prime Puzzles and Problems Connection.
%H A076335 Carlos Rivera, <a href="http://www.primepuzzles.net/private/index.htm">See here for latest information about progress on this sequence</a>
%H A076335 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrierNumber.html">Brier Number</a>
%Y A076335 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.
%Y A076335 A180247 gives the primes.
%Y A076335 See also A076336, A076337.
%Y A076335 A234594 is the old, incorrect version.
%K A076335 nonn
%O A076335 1,1
%A A076335 _Olivier Gérard_, Nov 07 2002
%E A076335 Many terms reported in Problem 29 from "The Prime Problems & Puzzles Connection" from _Carlos Rivera_, May 30 2010
%E A076335 Entry revised by _Arkadiusz Wesolowski_, May 17 2012
%E A076335 Entry revised by _Carlos Rivera_ and _N. J. A. Sloane_, Jan 03 2014
%E A076335 Entry revised by _Arkadiusz Wesolowski_, Feb 15 2014