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 A072276 Strong pseudoprimes to bases 2 and 3. 7

%I

%S 1373653,1530787,1987021,2284453,3116107,5173601,6787327,11541307,

%T 13694761,15978007,16070429,16879501,25326001,27509653,27664033,

%U 28527049,54029741,61832377,66096253,74927161,80375707,101649241

%N Strong pseudoprimes to bases 2 and 3.

%C Composites that pass the Miller-Rabin test for bases 2 and 3. The intersection of A001262 (strong pseudoprimes to base 2) and A020229 (strong pseudoprimes to base 3).

%C The Washington Bomfim link references a table with all terms up to 2^64. Data from Jan Feitsma and William Galway, see link below, permitted an easy determination of these terms. I tested the Mathematica function PrimeQ[n] with those numbers to verify that it is correct for all n < 2^64. - _Washington Bomfim_, May 13 2012

%H Don Reble, <a href="/A072276/b072276.txt">Table of n, a(n) for n = 1..10000</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.10, pp. 786-792

%H D. Bleichenbacher, <a href="http://www.bell-labs.com/user/bleichen/diss/thesis.html">Thesis and strong pseudoprimes to 2 and 3 up to 10^16</a>

%H Washington Bomfim, <a href="http://dl.dropbox.com/u/79527920/bF23.txt">Table of n, a(n) for n=1..1499371</a> [a large file]

%H Jan Feitsma and William Galway, <a href="http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html">Tables of pseudoprimes and related data</a>

%H A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, <a href="http://www.cacr.math.uwaterloo.ca/hac/">Handbook of Applied Cryptography</a>, section 4.2.3, Miller-Rabin test.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html">Rabin-Miller test</a>

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%t nmax = 10^8; sppQ[n_?EvenQ, _] := False; sppQ[n_?PrimeQ, _] := False; sppQ[n_, b_] := (s = IntegerExponent[n - 1, 2]; d = (n - 1)/2^s; If[ PowerMod[b, d, n] == 1, Return[True], Do[ If[ PowerMod[b, d*2^r, n] == n-1, Return[True]], {r, 0, s-1}]]); A072276 = {}; n = 1; While[n < nmax, n = n+2; If[sppQ[n, 2] && sppQ[n, 3] , Print[n]; AppendTo[ A072276, n]]]; A072276 (* _Jean-François Alcover_, Oct 20 2011, after _R. J. Mathar_ *)

%Y Cf. A001262, A006945, A014233, A020229.

%K nonn

%O 1,1

%A Francois Grieu (fgrieu(AT)micronet.fr), Jul 09 2002

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Last modified August 22 12:10 EDT 2019. Contains 326177 sequences. (Running on oeis4.)