|
%I M5362 #70 Feb 09 2022 01:03:11
%S 91,121,286,671,703,949,1105,1541,1729,1891,2465,2665,2701,2821,3281,
%T 3367,3751,4961,5551,6601,7381,8401,8911,10585,11011,12403,14383,
%U 15203,15457,15841,16471,16531,18721,19345,23521,24046,24661,24727,28009,29161
%N Pseudoprimes to base 3.
%C Theorem: If q>3 and both numbers q and (2q-1) are primes then n=q*(2q-1) is a pseudoprime to base 3 (i.e. n is in the sequence). So for n>2, A005382(n)*(2*A005382(n)-1) is in the sequence (see Comments lines for the sequence A122780). 91,703,1891,2701,12403,18721,38503,49141... are such terms. This sequence is a subsequence of A122780. - _Farideh Firoozbakht_, Sep 13 2006
%C Composite numbers n such that 3^(n-1) == 1 (mod n).
%C Theorem (R. Steuerwald, 1948): if n is a pseudoprime to base b and gcd(n,b-1)=1, then (b^n-1)/(b-1) is a pseudoprime to base b. In particular, if n is an odd pseudoprime to base 3, then (3^n-1)/2 is a pseudoprime to base 3. - _Thomas Ordowski_, Apr 06 2016
%C Steuerwald's theorem can be strengthened by weakening his assumption as follows: if n is a weak pseudoprime to base b and gcd(n,b-1)=1, then ... - _Thomas Ordowski_, Feb 23 2021
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 91, p. 33, Ellipses, Paris 2008.
%D R. K. Guy, Unsolved Problems in Number Theory, A12.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, <a href="/A005935/b005935.txt">Table of n, a(n) for n = 1..102839</a> (terms a(1)-a(798) from R. J. Mathar, a(799)-a(1000) from T. D. Noe)
%H J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/PSP.pdf">Pseudoprimes (Text in German)</a>
%H C. Pomerance & N. J. A. Sloane, <a href="/A001567/a001567_4.pdf">Correspondence, 1991</a>
%H F. Richman, <a href="http://math.fau.edu/Richman/carm.htm">Primality testing with Fermat's little theorem</a>
%H Rudolf Steuerwald, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1948_0069-0070.pdf">Über die Kongruenz a^(n-1) == 1 (mod n)</a>, Sitzungsber. math.-naturw. Kl. Bayer. Akad. Wiss. München, 1948, pp. 69-70.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>
%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%t base = 3; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* _T. D. Noe_, Feb 21 2012 *)
%o (PARI) is_A005935(n)={Mod(3,n)^(n-1)==1 & !ispseudoprime(n) & n>1} \\ _M. F. Hasler_, Jul 19 2012
%Y Pseudoprimes to other bases: A001567 (2), A005936 (5), A005937 (6), A005938 (7), A005939 (10).
%Y Subsequence of A122780.
%Y Cf. A005382.
%K nonn
%O 1,1
%A _N. J. A. Sloane_.
%E More terms from _David W. Wilson_, Aug 15 1996
|
