A072276 Strong pseudoprimes to bases 2 and 3.

1373653, 1530787, 1987021, 2284453, 3116107, 5173601, 6787327, 11541307, 13694761, 15978007, 16070429, 16879501, 25326001, 27509653, 27664033, 28527049, 54029741, 61832377, 66096253, 74927161, 80375707, 101649241

Offset: 1

Author

Francois R. Grieu, Jul 09 2002

Keywords

• nonn

Comments

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).

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

Links

• Don Reble, Table of n, a(n) for n = 1..10000
• Joerg Arndt, Matters Computational (The Fxtbook), section 39.10, pp. 786-792
• D. Bleichenbacher, Thesis and strong pseudoprimes to 2 and 3 up to 10^16
• Washington Bomfim, Table of n, a(n) for n=1..1499371 [a large file]
• Jan Feitsma and William Galway, Tables of pseudoprimes and related data
• A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, section 4.2.3, Miller-Rabin test.
• Eric Weisstein's World of Mathematics, Rabin-Miller test
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A001262, A006945, A014233, A020229.

Programs

Mathematica

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 *)