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A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p*2^k + 1 and p*2^k - 1 are composite.

%I A180247 #83 Feb 16 2025 08:33:13
%S A180247 10439679896374780276373,21444598169181578466233,
%T A180247 105404490005793363299729,178328409866851219182953,
%U A180247 239365215362656954573813,378418904967987321998467,422280395899865397194393,474362792344501650476113,490393518369132405769309
%N A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p*2^k + 1 and p*2^k - 1 are composite.
%C A180247 WARNING: These are just the smallest examples known - there may be smaller ones. Even the first term is uncertain. - _N. J. A. Sloane_, Jun 20 2017
%C A180247 There are no prime Brier numbers below 10^10. - _Arkadiusz Wesolowski_, Jan 12 2011
%C A180247 It is a conjecture that every such number has more than 11 digits. In 2011 I have calculated that for any prime p < 10^11 there is a k such that either p*2^k + 1 or p*2^k - 1 has all its prime factors greater than 1321. - _Arkadiusz Wesolowski_, Feb 03 2016
%C A180247 The first term was found by Dan Ismailescu and Peter Seho Park and the next two by Christophe Clavier (see below). See also A076335. - _N. J. A. Sloane_, Jan 03 2014
%C A180247 a(4)-a(9) computed in 2017 by the author.
%H A180247 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 A180247 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/RieselNumber.html">Riesel number</a>
%H A180247 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/SierpinskiNumber.html">Sierpinski number</a>
%H A180247 Christophe Clavier, <a href="/A076335/a076335.txt">14 new Brier numbers</a>
%H A180247 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 A180247 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 A180247 Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf">A search for some small Brier numbers</a>, 2000.
%H A180247 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/23122.html">Prime Curios! 6992565235279559197457863</a>
%H A180247 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 A180247 Joe McLean, <a href="http://oeis.org/A076336/a076336b.html">Brier Numbers</a> [Cached copy]
%H A180247 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_052.htm">Problem 52. ±p ± 2^n</a>, The Prime Puzzles and Problems Connection.
%H A180247 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrierNumber.html">Brier Number</a>
%Y A180247 Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639, A076336, A076337, A040081, A040076, A103963, A103964, A038699, A050921, A064699, A052333, A003261.
%Y A180247 These are the primes in A076335.
%K A180247 nonn
%O A180247 1,1
%A A180247 _Arkadiusz Wesolowski_, Aug 19 2010
%E A180247 Entry revised by _N. J. A. Sloane_, Jan 03 2014
%E A180247 Entry revised by _Arkadiusz Wesolowski_, May 29 2017